Unobserved Confounding,Bounds, and Sensitivity Analysis
Brady Nealcausalcourse.com

Motivation: Unobserved Confounding<latexit sha1_base64="uMTSYG5prIZDfUs1Kq56hBXH/i0=">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</latexit>

W

T

E[Y (1)<latexit sha1_base64="qEU0nKhHJfNQr73O6QeY2sLiLXE=">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</latexit><latexit sha1_base64="qEU0nKhHJfNQr73O6QeY2sLiLXE=">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</latexit>

Brady Neal

Y (0)] = EW [E[Y | T = 1, W ]Y

E[Y | T = 0, W ]]2 / 48

Motivation: Unobserved Confounding<latexit sha1_base64="9eM2YXHVVZWuQrh44ZrzxH0Pvyo=">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</latexit>

E[Y (1)<latexit sha1_base64="qEU0nKhHJfNQr73O6QeY2sLiLXE=">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</latexit><latexit sha1_base64="7h4SIYwEFBbP5TqSm2XIubJIUPU=">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</latexit>

Brady Neal

W

U

T

Y

WW,
] UE[Y|T =W0,]] W, U ]]Y (0)] = EW,U
[E[Y| T| T==1,1,] E[Y| T0,=
W [E[Y
2 / 48

Motivation: Unobserved Confounding<latexit sha1_base64="9eM2YXHVVZWuQrh44ZrzxH0Pvyo=">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</latexit>

E[Y (1)<latexit sha1_base64="qEU0nKhHJfNQr73O6QeY2sLiLXE=">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</latexit><latexit sha1_base64="7h4SIYwEFBbP5TqSm2XIubJIUPU=">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</latexit>

W

U

T

Y

WW,
] UE[Y|T =W0,]] W, U ]]Y (0)] = EW,U
[E[Y| T| T==1,1,] E[Y| T0,=
W [E[Y<latexit sha1_base64="vei1mn9NO6vHWjVzBryobX//Eg0=">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</latexit>?

⇡ EW [E[Y | T = 1, W ]Brady NealE[Y | T = 0, W ]]2 / 48

Bounds
No-Assumptions Bound
Monotone Treatment Response
Monotone Treatment Selection
Optimal Treatment Selection

Sensitivity Analysis
Linear Single Confounder
Towards More General Settings

Brady Neal

3 / 48

Bounds
No-Assumptions Bound
Monotone Treatment Response
Monotone Treatment Selection
Optimal Treatment Selection

Sensitivity Analysis
Linear Single Confounder
Towards More General Settings

Brady Neal

Bounds4 / 48

“No Unobserved Confounding" is Unrealistic

Brady Neal

Bounds

5 / 48

“No Unobserved Confounding" is Unrealistic“The Law of Decreasing Credibility: The credibility of inference decreases
with the strength of the assumptions maintained” (Manski, 2003).

Brady Neal

Bounds

5 / 48

“No Unobserved Confounding" is Unrealistic“The Law of Decreasing Credibility: The credibility of inference decreases
with the strength of the assumptions maintained” (Manski, 2003).
Assume unconfoundedness. Then,
E[Y (1)<latexit sha1_base64="Cd0ejn/2IrgA2vVI5STy7lQeO30=">AAAUTnicrVjbbttGEF2nN8e9OXHfChRsnKIOIKuSmyJ5MRDAcVAUDZAUzq2WYUgkZRHmrSQV1xH0E/2yvvZHir4U7ZmzS5HUjXIQCRJXszNnZue2u+rFvpdmrdZfa9fee/+DDz9av77x8Seffvb55o2bz9NomNjuMzvyo+Rlr5u6vhe6zzIv892XceJ2g57vvuidH8j8i9duknpReJRdxu5J0D0Lvb5ndzOQTjf/6Bwev9pp37F2rVc7rTsn1r7VOTwddewo7EfD0IHo2Or4bj87xsRxJxpmdhS4VifwHKuTQVUWuGEGsXbDKkudAHG5QGtaoJN4Z4Ps5HRzu9Vs8WXNDtpmsK3M60l04/ovqqMcFSlbDVWgXBWqDGNfdVWK97Fqq5aKQTtRI9ASjDzOu2qsNiA7BJcLji6o5/g+w69jQw3xWzBTStvQ4uOTQNJS3xgeB+M+qfop+q0S7yIdI2KLjZd49gxmAGqmBqDWyeWcq8v18A5qVp1hDfe5Wg8riUkRP9iVNffx9PE7wwrl+xKcLkYOpBKMbNB8UDVFdCR4as+LbwaMRJd8LkZi9TK7l1kt8xHe58DqYpzSUrHVUo9MXEJqdmmr8PiM6WLE37FCbd8yrP5kDR7jrlchvEegnKs3GFWRl+ksZ9dirswgj+foEv6MHCG4U1AiZr4H/3ngqOaqDUzR0WUkz7iWmBnUNMg/cT7P5hjx3KU9KaNmMd886olNZRV253b6ePaInUB+hLkB9YgvGrDHJXbC7M292CB3gl8X/GXTRnuK3mRlSjwbWJNkVwO4EWa8CZb2hPZnbpG2bgSaZd676mfqdxH3Br0ied2AbEQLdRan1BWa2O+jr4hfAnwLVTybUxrUEdCOAbPlHLRCn0YSfd9hFSnrxzeoIzx9ei0mSoPaxT8XHAdck0R0SN0XGDvUI9IWel1T3TN2jCtaxTceu9OsVov5J7WrO+O0VpELWamWycp9amqpu2bFEgGxu+w123RD3ZNTrm48ZVWP9hddQ3K0ap9IFfGpxsphNg4oVcUVL85f57y17LFOGoyl+P4MHPuU6ht6ytqMjaaNUv0dmCqJGA+bz8DkhK530XYxNTMCPTCe8sGn61z2JJn5DZhd5lSH3ivPL0M8NPL5biAdf0TqcrnnrKyqbILRaDKzXN7jyKV/9EgirvFi+jrmWr6efKzSzKrY7wYvnEGUjM0M0qroDnNj/sqP1DZ6wTayqupPibRQEp5OYoxvTzhvA1d4z6klBEUqfDSZH9fG4CEoY+OxPufyOixs8Nj5HHJ2TOZpDl1R2RSHVPVyrZKX8/JG05fLOpN8r8pq+nJZobxmbnqmH6TEelkjlzEOIhGUsvSoRirinqDp2uZXK9gXcq8dmrxIjIUvaiQD9hi9qshU5eNa+zJ2wcT4RGR+fQsPvmb/G/MEdFU/FrLZlbxZyF2+lU8L+YsreraQDK7o30LyTY2ky73Bm9BE5mltTQnXk1ounU91e4KL3SQ/v+hqa/C7By/Knu2YzmSZ/UayQu+O++wBjZX68+y+I5Xcrq3jIbtvj30ooWZnSe08e2doRQSHZi/Pd/P8FJ6W+rxHjcUJfcTT1IXZEfJz1S49qX3xI3rKY5xVZHygvsc5qY3vR5XdY1VUua8MTd8p4+4B+R76/kPuD1fHjc2tYrdy96jqOISG/DM2t7SC2zdn6UVIV9E7H1lHz5k5009r0qe2Abj0TeayBrHIh/l4y+M7H1n/4xDxJF3cZ94FcvmuthpekTOLb9Z19/3Ft9AYzz6+fXbG1PA+ZNb57JAualXflPUp5GDmzi/5u4eqkNwd3bo5pa+KP32LPOMtcAiOgkffBjJKa1oZcdF9NKZMZk5dKe9uxQ21Wblxarxp29JSXgUGb990KNd03Y3Tze329L9Ys4Pne832D83W07vbD+6bf7jW1ZfqltqBn+6pB6jMJ/Crrf5Z+2rt27WdrT+3/t76d+s/zXptzchsqcrri/X/ARgEAmw=</latexit>

Brady Neal

Y (0)] = EW [E[Y | T = 1, W ]BoundsE[Y | T = 0, W ]]5 / 48

“No Unobserved Confounding" is Unrealistic“The Law of Decreasing Credibility: The credibility of inference decreases
with the strength of the assumptions maintained” (Manski, 2003).
Assume unconfoundedness. Then,
E[Y (1)<latexit sha1_base64="Cd0ejn/2IrgA2vVI5STy7lQeO30=">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</latexit>

Y (0)] = EW [E[Y | T = 1, W ]Identify a point

Brady Neal

BoundsE[Y | T = 0, W ]]5 / 48

“No Unobserved Confounding" is Unrealistic“The Law of Decreasing Credibility: The credibility of inference decreases
with the strength of the assumptions maintained” (Manski, 2003).
Assume unconfoundedness. Then,
E[Y (1)<latexit sha1_base64="Cd0ejn/2IrgA2vVI5STy7lQeO30=">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</latexit>

Y (0)] = EW [E[Y | T = 1, W ]Identify a point

E[Y | T = 0, W ]]Make weaker assumptions:

Brady Neal

Bounds

5 / 48

“No Unobserved Confounding" is Unrealistic“The Law of Decreasing Credibility: The credibility of inference decreases
with the strength of the assumptions maintained” (Manski, 2003).
Assume unconfoundedness. Then,
E[Y (1)<latexit sha1_base64="Cd0ejn/2IrgA2vVI5STy7lQeO30=">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</latexit>

Y (0)] = EW [E[Y | T = 1, W ]Identify a point

E[Y | T = 0, W ]]Make weaker assumptions:
Identify an interval
Brady Neal

Bounds

5 / 48

“No Unobserved Confounding" is Unrealistic“The Law of Decreasing Credibility: The credibility of inference decreases
with the strength of the assumptions maintained” (Manski, 2003).
Assume unconfoundedness. Then,
E[Y (1)<latexit sha1_base64="Cd0ejn/2IrgA2vVI5STy7lQeO30=">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</latexit>

Y (0)] = EW [E[Y | T = 1, W ]Identify a point

E[Y | T = 0, W ]]Make weaker assumptions:
Identify an interval“Partial identification” or “set identification”
Brady Neal

Bounds

5 / 48

Bounds
No-Assumptions Bound
Monotone Treatment Response
Monotone Treatment Selection
Optimal Treatment Selection

Sensitivity Analysis
Linear Single Confounder
Towards More General Settings

Brady Neal

Bounds: No-Assumptions Bound

6 / 48

Bounded Potential Outcomes

Brady Neal

Bounds: No-Assumptions Bound

7 / 48

Bounded Potential Outcomes
Example: Y(0) and Y(1) are between 0 and 1

Brady Neal

Bounds: No-Assumptions Bound

7 / 48

Bounded Potential Outcomes
Example: Y(0) and Y(1) are between 0 and 1<latexit sha1_base64="KCeIucixt9He3yDyS4naIz42+8c=">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</latexit>

Brady Neal

1  Yi (1)Yi (0)  1

Bounds: No-Assumptions Bound

Max: 1 - 0

7 / 48

Bounded Potential Outcomes
Example: Y(0) and Y(1) are between 0 and 1
1  Yi (1) Yi (0)  1
Min: 0 - 1<latexit sha1_base64="KCeIucixt9He3yDyS4naIz42+8c=">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</latexit><latexit sha1_base64="KCeIucixt9He3yDyS4naIz42+8c=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

Max: 1 - 0

7 / 48

Bounded Potential Outcomes
Example: Y(0) and Y(1) are between 0 and 1
1  Yi (1) Yi (0)  1
Min: 0 - 1<latexit 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Brady Neal

1  E[Y (1)Max: 1 - 0

Y (0)]  1

Bounds: No-Assumptions Bound

7 / 48

Bounded Potential Outcomes
Example: Y(0) and Y(1) are between 0 and 1
1  Yi (1) Yi (0)  1
Min: 0 - 1<latexit 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1  E[Y (1)Max: 1 - 0

Y (0)]  1

Trivial length limit: 2

Brady Neal

Bounds: No-Assumptions Bound

7 / 48

Bounded Potential Outcomes
Example: Y(0) and Y(1) are between 0 and 1
1  Yi (1) Yi (0)  1
Min: 0 - 1<latexit 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Spoiler: We can cut this in half

Brady Neal

1  E[Y (1)Max: 1 - 0

Y (0)]  1

Trivial length limit: 2

Bounds: No-Assumptions Bound

7 / 48

Bounded Potential Outcomes
Example: Y(0) and Y(1) are between 0 and 1
1  Yi (1) Yi (0)  1
Min: 0 - 1<latexit 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Spoiler: We can cut this in half

1  E[Y (1)Max: 1 - 0

Y (0)]  1

Trivial length limit: 2More generally, potential outcomes are bounded: 8t, a  Y (t)  b<latexit sha1_base64="BRynu6VyKGRrzeKMhOG3InQ+E3E=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

7 / 48

Bounded Potential Outcomes
Example: Y(0) and Y(1) are between 0 and 1
1  Yi (1) Yi (0)  1
Min: 0 - 1<latexit 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Spoiler: We can cut this in half

1  E[Y (1)Max: 1 - 0

Y (0)]  1

Trivial length limit: 2More generally, potential outcomes are bounded: 8t, a  Y (t)  b<latexit sha1_base64="BRynu6VyKGRrzeKMhOG3InQ+E3E=">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</latexit><latexit sha1_base64="Y5iTbP+KPwcQ56A4a+bCloJ9+fQ=">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</latexit>a

Brady Nealb  Yi (1)Yi (0)  b

Bounds: No-Assumptions Bounda

7 / 48

Bounded Potential Outcomes
Example: Y(0) and Y(1) are between 0 and 1
1  Yi (1) Yi (0)  1
Min: 0 - 1<latexit 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Spoiler: We can cut this in half

1  E[Y (1)Max: 1 - 0

Y (0)]  1

Trivial length limit: 2More generally, potential outcomes are bounded: 8t, a  Y (t)  b<latexit 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Brady Nealb  Yi (1)b  E[Y (1)Yi (0)  b

Y (0)]  b

Bounds: No-Assumptions Boundaa

7 / 48

Bounded Potential Outcomes
Example: Y(0) and Y(1) are between 0 and 1
1  Yi (1) Yi (0)  1
Min: 0 - 1<latexit 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Spoiler: We can cut this in half

1  E[Y (1)Max: 1 - 0

Y (0)]  1

Trivial length limit: 2More generally, potential outcomes are bounded: 8t, a  Y (t)  b<latexit 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 Yi (1)b  E[Y (1)Yi (0)  ba

Y (0)]  baTrivial length limit: 2(ba)<latexit sha1_base64="xDUWc/MYNypUpJO9sx67/fWnYkY=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

7 / 48

Bounded Potential Outcomes
Example: Y(0) and Y(1) are between 0 and 1
1  Yi (1) Yi (0)  1
Min: 0 - 1<latexit 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Spoiler: We can cut this in half

1  E[Y (1)Max: 1 - 0

Y (0)]  1

Trivial length limit: 2More generally, potential outcomes are bounded: 8t, a  Y (t)  b<latexit 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sha1_base64="yhX9WtXCdiGLM3RE0rncm1vlvfw=">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</latexit>a

Spoiler: We can cut this in half
Brady Nealb  Yi (1)b  E[Y (1)Yi (0)  ba

Y (0)]  baTrivial length limit: 2(ba)<latexit sha1_base64="xDUWc/MYNypUpJO9sx67/fWnYkY=">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</latexit>

Bounds: No-Assumptions Bound

7 / 48Observational-Counterfactual Decomposition

Proof:<latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit>

E[Y (1)

Y (0)] = E[Y (1)]

E[Y (0)]= P (T = 1) E[Y (1) | T = 1] + P (T = 0) E[Y (1) | T = 0]
P (T = 1) E[Y (0) | T = 1]

P (T = 0) E[Y (0) | T = 0]

P (T = 1) E[Y (0) | T = 1]

P (T = 0) E[Y | T = 0]

= P (T = 1) E[Y | T = 1] + P (T = 0) E[Y (1) | T = 0]
Brady Neal

Bounds: No-Assumptions Bound

8 / 48Observational-Counterfactual Decomposition

Proof:<latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit><latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit>

E[Y (1)

Y (0)] = E[Y (1)] E[Y (0)]= P (T = 1) E[Y (1) | T = 1] + P (T = 0) E[Y (1) | T = 0]

P (T = 1) E[Y (0) | T = 1] P (T = 0) E[Y (0) | T = 0]
= P (T = 1) E[Y | T = 1] + P (T = 0) E[Y (1) | T = 0]
P (T = 1) E[Y (0) | T = 1] P (T = 0) E[Y | T = 0]

Brady Neal

Bounds: No-Assumptions Bound

8 / 48Observational-Counterfactual Decomposition

Proof:<latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit><latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit><latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">AAAWGXic3VjLjhtFFK2YVzK8kqBZsWkyASXCY+whKNlYijRJhBCREjR5QDyK7O72TGv6le52honlL+FrEBvElhWfwBZ+gHNPVbu7/Wo7ygpbdpdv3XvurfuqKg9i30uzdvuvc4233n7n3ffOX9h6/4MPP/r44qXLj9NolNjuIzvyo+TpoJ+6vhe6jzIv892nceL2g4HvPhmc7Mv8k5duknpReJCdxe5h0D8KvaFn9zOQnl9q9Hp3n/WiUWZHgXutc93ataa/2tcPrS+6VpXhUDhKFGHq9bYsvIQ3vtbLoD8L3DCzuhYAe80ZBKsXeI5VZTu0vpyTba8n24b+MApHwcBNppb0Xoz6jlhaa097iT3zsvP2LJRtb+SP/40zFhrz/OJOu9Xmy5ofdMxgR5nXg+jShR9UTzkqUrYaqUC5KlQZxr7qqxTvZ6qj2ioG7VCNQUsw8jjvqonaguwIXC44+qCe4PsIv54ZaojfgplS2oYWH58Ekpb63PA4GA9J1U/Rb5V4l+kYE1tsPMNzYDADUDN1DGqdXM65vtwA76Bm1RnWcIur9bCSmBTxg11Z8xBPH78zrFC+z8DpYuRAKsHIBs0HVVNER4Kn9rz45piR6JPPxUisXmX3KqtlPsL7BFh9jFNaKrZa6p6JS0jNLm0VHp8xXY74M1ao7VuFNZyuwWPc9SqE9wCUE/UKoyryKp3l7FrOlRnkyQJdwp+RIwR3CkrEzPfgPw8c1Vy1gSk6+ozkEdcSM4NaBvk7zufZHCOeu7QnZdQs5ptHPbGprMLu3E4fzwGxE8iPMXdMPeKLJuxxiZ0we3MvNsmd4Ncpf9m00Z6ht1iZEs8m1iTZ1QRuhBlviqU9of2ZW6StG4Nmmfeu+p76XcS9Sa9IXjchG9FCncUpdYUm9l30FfFLgG+himdzSpM6AtpxzGw5Aa3Qp5FE31dYRcr68Q3qGE+fXouJ0qR28c8pxwHXJBEdUfcpxg71iLSFXtdSN40dk4pW8Y3H7jSv1WL+Se3qzjirVeRCVqplsrJLTW11w6xYIiB2l71mm26oe3LK1U1mrBrQ/qJrSI5W7ROpIj7VWDnMxmNKVXHFi4vXuWgte6yTJmMpvj8CR5dSQ0NPWZux0bRVqr99UyUR42HzGZic0PUu2k5nZsagB8ZTPvh0ncueJDMvgNlnTvXovfL8KsS7Rj7fDaTjj0ldLfeYlVWVTTAaT2dWy3scufSPHknENV5MX8dcy2fTj1WaWRf7zeCFc4iSsZlBWhfdYW4sXvmB2kEv2EFWVf0pkRZKwtNJjPHVKedV4ArvCbWEoEiFj6fzk9oY3AFlYjw25Fxeh4UNHjufQ86eyTzNoSsqm+GQql6tVfJyUd5o+mpZZ5rvVVlNXy0rlJfMTc/0g5RYT2vkMsZBJIJSlh7USEXcEzRd2/zjGvaF3GtHJi8SY+GTGsmAPUavKjJVeb/WvoxdMDE+EZmfXsODL9n/JjwBberHQjbbyJuF3Nlr+bSQP93Qs4VksKF/C8lXNZIu9wZvShOZh7U1JVwParl0PtXtCS52k/z8oqutye8BvCh7tmM6k2X2G8kKvTt22QOaa/Xn+X1HKrlTW8cjdt8B+1BCzc6K2nn0xtCKCI7MXp7v5vkpPC31eY8aixP6mKepU7Mj5OeqXXpS++Jb9JT7OKvIeF99jXNSB9/3KrvHuqhyXxmZvlPG3QPyTfT9O9wfNseNza1it3L3qOq4Cw35Z2JuaQW3b87Sy5A20bsYWUfPmTvTz2rSp7ZjcOmbzFkNYpEPi/FWx3cxsv7HIeJJurjPvAnk8l1tPbwiZ5bfrOvu+8tvoTGeQ3z77Iyp4b3DrPPZIV3Uqr4p61PI/tydX/J3D1UhuTu+cnlGXxV/9hZ5xFvgCBwFj74NZJTWtDLisvtoTJnMnLpS3t2KG2qrcuPUeLO2paW8Cgxe13Qo13TdrecXdzqz/2LNDx7vtTrftNoPb+zcvmX+4TqvPlVX1DX46aa6jcp8AL/ajd8afzf+afy7/cv2r9u/b/+hWRvnjMwnqvLa/vM/CqGQcg==</latexit>

E[Y(1)
E[Y (1)= E[YE[Y (1)](1)] E[YE[Y (0)](0)]
Y (0)](0)] =
Y
=P
P (T
(T =
= 1)
1) E[YE[Y (1)(1) || T
T =
= 1]
1] +
+P
P (T
(T =
= 0)
0) E[Y=E[Y (1)(1) || T
T =
= 0]
0]
P (T
(T =
= 1)
1) E[YE[Y (0)(0) || T
T =
= 1]
1] P
P (T
(T =
= 0)
P
0) E[YE[Y (0)(0) || T
T =
= 0]
0]
=P
P (T
(T =
= 1)
1) E[YE[Y || T
T =
= 1]
1] +
+P
P (T
(T =
= 0)
0) E[YE[Y (1)=(1) || T
T =
= 0]
0]
P (T
(T =
= 1)
1) E[YE[Y (0)(0) || T
T =
= 1]
1] P
P (T
(T =
= 0)
P
0) E[YE[Y || T
T =
= 0]
0]

Brady Neal

Bounds: No-Assumptions Bound

8 / 48Observational-Counterfactual Decomposition

Proof:<latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">AAAWGXic3VjLjhtFFK2YVzK8kqBZsWkyASXCY+whKNlYijRJhBCREjR5QDyK7O72TGv6le52honlL+FrEBvElhWfwBZ+gHNPVbu7/Wo7ygpbdpdv3XvurfuqKg9i30uzdvuvc4233n7n3ffOX9h6/4MPP/r44qXLj9NolNjuIzvyo+TpoJ+6vhe6jzIv892nceL2g4HvPhmc7Mv8k5duknpReJCdxe5h0D8KvaFn9zOQnl9q9Hp3n/WiUWZHgXutc93ataa/2tcPrS+6VpXhUDhKFGHq9bYsvIQ3vtbLoD8L3DCzuhYAe80ZBKsXeI5VZTu0vpyTba8n24b+MApHwcBNppb0Xoz6jlhaa097iT3zsvP2LJRtb+SP/40zFhrz/OJOu9Xmy5ofdMxgR5nXg+jShR9UTzkqUrYaqUC5KlQZxr7qqxTvZ6qj2ioG7VCNQUsw8jjvqonaguwIXC44+qCe4PsIv54ZaojfgplS2oYWH58Ekpb63PA4GA9J1U/Rb5V4l+kYE1tsPMNzYDADUDN1DGqdXM65vtwA76Bm1RnWcIur9bCSmBTxg11Z8xBPH78zrFC+z8DpYuRAKsHIBs0HVVNER4Kn9rz45piR6JPPxUisXmX3KqtlPsL7BFh9jFNaKrZa6p6JS0jNLm0VHp8xXY74M1ao7VuFNZyuwWPc9SqE9wCUE/UKoyryKp3l7FrOlRnkyQJdwp+RIwR3CkrEzPfgPw8c1Vy1gSk6+ozkEdcSM4NaBvk7zufZHCOeu7QnZdQs5ptHPbGprMLu3E4fzwGxE8iPMXdMPeKLJuxxiZ0we3MvNsmd4Ncpf9m00Z6ht1iZEs8m1iTZ1QRuhBlviqU9of2ZW6StG4Nmmfeu+p76XcS9Sa9IXjchG9FCncUpdYUm9l30FfFLgG+himdzSpM6AtpxzGw5Aa3Qp5FE31dYRcr68Q3qGE+fXouJ0qR28c8pxwHXJBEdUfcpxg71iLSFXtdSN40dk4pW8Y3H7jSv1WL+Se3qzjirVeRCVqplsrJLTW11w6xYIiB2l71mm26oe3LK1U1mrBrQ/qJrSI5W7ROpIj7VWDnMxmNKVXHFi4vXuWgte6yTJmMpvj8CR5dSQ0NPWZux0bRVqr99UyUR42HzGZic0PUu2k5nZsagB8ZTPvh0ncueJDMvgNlnTvXovfL8KsS7Rj7fDaTjj0ldLfeYlVWVTTAaT2dWy3scufSPHknENV5MX8dcy2fTj1WaWRf7zeCFc4iSsZlBWhfdYW4sXvmB2kEv2EFWVf0pkRZKwtNJjPHVKedV4ArvCbWEoEiFj6fzk9oY3AFlYjw25Fxeh4UNHjufQ86eyTzNoSsqm+GQql6tVfJyUd5o+mpZZ5rvVVlNXy0rlJfMTc/0g5RYT2vkMsZBJIJSlh7USEXcEzRd2/zjGvaF3GtHJi8SY+GTGsmAPUavKjJVeb/WvoxdMDE+EZmfXsODL9n/JjwBberHQjbbyJuF3Nlr+bSQP93Qs4VksKF/C8lXNZIu9wZvShOZh7U1JVwParl0PtXtCS52k/z8oqutye8BvCh7tmM6k2X2G8kKvTt22QOaa/Xn+X1HKrlTW8cjdt8B+1BCzc6K2nn0xtCKCI7MXp7v5vkpPC31eY8aixP6mKepU7Mj5OeqXXpS++Jb9JT7OKvIeF99jXNSB9/3KrvHuqhyXxmZvlPG3QPyTfT9O9wfNseNza1it3L3qOq4Cw35Z2JuaQW3b87Sy5A20bsYWUfPmTvTz2rSp7ZjcOmbzFkNYpEPi/FWx3cxsv7HIeJJurjPvAnk8l1tPbwiZ5bfrOvu+8tvoTGeQ3z77Iyp4b3DrPPZIV3Uqr4p61PI/tydX/J3D1UhuTu+cnlGXxV/9hZ5xFvgCBwFj74NZJTWtDLisvtoTJnMnLpS3t2KG2qrcuPUeLO2paW8Cgxe13Qo13TdrecXdzqz/2LNDx7vtTrftNoPb+zcvmX+4TqvPlVX1DX46aa6jcp8AL/ajd8afzf+afy7/cv2r9u/b/+hWRvnjMwnqvLa/vM/CqGQcg==</latexit><latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit><latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit><latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit>

E[Y
(1)
E[Y
E[Y (1)(1)

Brady Neal

= E[Y (1)](1)] E[Y (0)](0)]
Y (0)] =Y
Y (0)](0)] = E[YE[Y (1)] E[YE[Y (0)]= P (T(T = 1)
1) E[Y (1)(1) | T == 1] ++ P (T(T = 0)0) E[Y (1)
|T =
0]
=
=P
P (T == 1) E[YE[Y (1) || T
T = 1]
1] + PP (T == 0) E[YE[Y (1)(1) || T
T =
= 0]
0]
P (T == 1) E[YE[Y (0) || T
T = 1]
1] P (T(T = 0)E[Y (0)
|T =
0]
P
P (T
(T = 1)
1) E[Y (0)(0) | T == 1] PP (T == 0)0) E[YE[Y (0)(0) || T
T =
= 0]
0]
=P
P (T
(T =
= 1)
1) E[YE[Y || T
T =
= 1]
1] +
+P
P (T
(T =
= 0)
0) E[YE[Y (1)
|| T
=
0]
=
(1)
T
=
= P (T = 1) E[Y | T = 1] + P (T = 0) E[Y (1) | T = 0]
0]
P (T
(T =
= 1)
1) E[YE[Y (0)(0) || T
T =
= 1]
1] P
P (T
(T =
= 0)
E[Y
|| T
=
0]
P
0)
E[Y
T
=
P (T = 1) E[Y (0) | T = 1] P (T = 0) E[Y | T = 0]0]Bounds: No-Assumptions Bound

8 / 48Observational-Counterfactual Decomposition

Proof:<latexit 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sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit><latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit><latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit>

E[Y
(1)
E[Y
E[Y (1)(1)

Brady Neal

= E[Y (1)](1)] E[Y (0)](0)]
Y (0)] =Y
Y (0)](0)] = E[YE[Y (1)] E[YE[Y (0)]= P (T(T = 1)
1) E[Y (1)(1) | T == 1] ++ P (T(T = 0)0) E[Y (1)
|T =
0]
=
=P
P (T == 1) E[YE[Y (1) || T
T = 1]
1] + PP (T == 0) E[YE[Y (1)(1) || T
T =
= 0]
0]
P (T == 1) E[YE[Y (0) || T
T = 1]
1] P (T(T = 0)E[Y (0)
|T =
0]
P
P (T
(T = 1)
1) E[Y (0)(0) | T == 1] PP (T == 0)0) E[YE[Y (0)(0) || T
T =
= 0]
0]
=P
P (T
(T =
= 1)
1) E[YE[Y || T
T =
= 1]
1] +
+P
P (T
(T =
= 0)
0) E[YE[Y (1)
|| T
=
0]
=
(1)
T
=
= P (T = 1) E[Y | T = 1] + P (T = 0) E[Y (1) | T = 0]
0]
P (T
(T =
= 1)
1) E[YE[Y (0)(0) || T
T =
= 1]
1] P
P (T
(T =
= 0)
E[Y
|| T
=
0]
P
0)
E[Y
T
=
P (T = 1) E[Y (0) | T = 1] P (T = 0) E[Y | T = 0]0]Bounds: No-Assumptions Bound

8 / 48Observational-Counterfactual Decomposition

Proof:<latexit 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sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit><latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit><latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit>

E[Y
(1)
E[Y
E[Y (1)(1)

Brady Neal

= E[Y (1)](1)] E[Y (0)](0)]
Y (0)] =Y
Y (0)](0)] = E[YE[Y (1)] E[YE[Y (0)]= P (T(T = 1)
1) E[Y (1)(1) | T == 1] ++ P (T(T = 0)0) E[Y (1)
|T =
0]
=
=P
P (T == 1) E[YE[Y (1) || T
T = 1]
1] + PP (T == 0) E[YE[Y (1)(1) || T
T =
= 0]
0]
P (T == 1) E[YE[Y (0) || T
T = 1]
1] P (T(T = 0)E[Y (0)
|T =
0]
P
P (T
(T = 1)
1) E[Y (0)(0) | T == 1] PP (T == 0)0) E[YE[Y (0)(0) || T
T =
= 0]
0]
=P
P (T
(T =
= 1)
1) E[YE[Y || T
T =
= 1]
1] +
+P
P (T
(T =
= 0)
0) E[YE[Y (1)
|| T
=
0]
=
(1)
T
=
= P (T = 1) E[Y | T = 1] + P (T = 0) E[Y (1) | T = 0]
0]
P (T
(T =
= 1)
1) E[YE[Y (0)(0) || T
T =
= 1]
1] P
P (T
(T =
= 0)
E[Y
|| T
=
0]
P
0)
E[Y
T
=
P (T = 1) E[Y (0) | T = 1] P (T = 0) E[Y | T = 0]0]Bounds: No-Assumptions Bound

8 / 48Observational-Counterfactual Decomposition

Proof:<latexit 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sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit><latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">AAAWGXic3VjLjhtFFK2YVzK8kqBZsWkyASXCY+whKNlYijRJhBCREjR5QDyK7O72TGv6le52honlL+FrEBvElhWfwBZ+gHNPVbu7/Wo7ygpbdpdv3XvurfuqKg9i30uzdvuvc4233n7n3ffOX9h6/4MPP/r44qXLj9NolNjuIzvyo+TpoJ+6vhe6jzIv892nceL2g4HvPhmc7Mv8k5duknpReJCdxe5h0D8KvaFn9zOQnl9q9Hp3n/WiUWZHgXutc93ataa/2tcPrS+6VpXhUDhKFGHq9bYsvIQ3vtbLoD8L3DCzuhYAe80ZBKsXeI5VZTu0vpyTba8n24b+MApHwcBNppb0Xoz6jlhaa097iT3zsvP2LJRtb+SP/40zFhrz/OJOu9Xmy5ofdMxgR5nXg+jShR9UTzkqUrYaqUC5KlQZxr7qqxTvZ6qj2ioG7VCNQUsw8jjvqonaguwIXC44+qCe4PsIv54ZaojfgplS2oYWH58Ekpb63PA4GA9J1U/Rb5V4l+kYE1tsPMNzYDADUDN1DGqdXM65vtwA76Bm1RnWcIur9bCSmBTxg11Z8xBPH78zrFC+z8DpYuRAKsHIBs0HVVNER4Kn9rz45piR6JPPxUisXmX3KqtlPsL7BFh9jFNaKrZa6p6JS0jNLm0VHp8xXY74M1ao7VuFNZyuwWPc9SqE9wCUE/UKoyryKp3l7FrOlRnkyQJdwp+RIwR3CkrEzPfgPw8c1Vy1gSk6+ozkEdcSM4NaBvk7zufZHCOeu7QnZdQs5ptHPbGprMLu3E4fzwGxE8iPMXdMPeKLJuxxiZ0we3MvNsmd4Ncpf9m00Z6ht1iZEs8m1iTZ1QRuhBlviqU9of2ZW6StG4Nmmfeu+p76XcS9Sa9IXjchG9FCncUpdYUm9l30FfFLgG+himdzSpM6AtpxzGw5Aa3Qp5FE31dYRcr68Q3qGE+fXouJ0qR28c8pxwHXJBEdUfcpxg71iLSFXtdSN40dk4pW8Y3H7jSv1WL+Se3qzjirVeRCVqplsrJLTW11w6xYIiB2l71mm26oe3LK1U1mrBrQ/qJrSI5W7ROpIj7VWDnMxmNKVXHFi4vXuWgte6yTJmMpvj8CR5dSQ0NPWZux0bRVqr99UyUR42HzGZic0PUu2k5nZsagB8ZTPvh0ncueJDMvgNlnTvXovfL8KsS7Rj7fDaTjj0ldLfeYlVWVTTAaT2dWy3scufSPHknENV5MX8dcy2fTj1WaWRf7zeCFc4iSsZlBWhfdYW4sXvmB2kEv2EFWVf0pkRZKwtNJjPHVKedV4ArvCbWEoEiFj6fzk9oY3AFlYjw25Fxeh4UNHjufQ86eyTzNoSsqm+GQql6tVfJyUd5o+mpZZ5rvVVlNXy0rlJfMTc/0g5RYT2vkMsZBJIJSlh7USEXcEzRd2/zjGvaF3GtHJi8SY+GTGsmAPUavKjJVeb/WvoxdMDE+EZmfXsODL9n/JjwBberHQjbbyJuF3Nlr+bSQP93Qs4VksKF/C8lXNZIu9wZvShOZh7U1JVwParl0PtXtCS52k/z8oqutye8BvCh7tmM6k2X2G8kKvTt22QOaa/Xn+X1HKrlTW8cjdt8B+1BCzc6K2nn0xtCKCI7MXp7v5vkpPC31eY8aixP6mKepU7Mj5OeqXXpS++Jb9JT7OKvIeF99jXNSB9/3KrvHuqhyXxmZvlPG3QPyTfT9O9wfNseNza1it3L3qOq4Cw35Z2JuaQW3b87Sy5A20bsYWUfPmTvTz2rSp7ZjcOmbzFkNYpEPi/FWx3cxsv7HIeJJurjPvAnk8l1tPbwiZ5bfrOvu+8tvoTGeQ3z77Iyp4b3DrPPZIV3Uqr4p61PI/tydX/J3D1UhuTu+cnlGXxV/9hZ5xFvgCBwFj74NZJTWtDLisvtoTJnMnLpS3t2KG2qrcuPUeLO2paW8Cgxe13Qo13TdrecXdzqz/2LNDx7vtTrftNoPb+zcvmX+4TqvPlVX1DX46aa6jcp8AL/ajd8afzf+afy7/cv2r9u/b/+hWRvnjMwnqvLa/vM/CqGQcg==</latexit><latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit>

E[Y
(1)
E[Y
E[Y (1)(1)= E[Y (1)](1)] E[Y (0)](0)]
Y (0)] =Y
Y (0)](0)] = E[YE[Y (1)] E[YE[Y (0)]= P (T(T = 1)
1) E[Y (1)(1) | T == 1] ++ P (T(T = 0)0) E[Y (1)
|T =
0]
=
=P
P (T == 1) E[YE[Y (1) || T
T = 1]
1] + PP (T == 0) E[YE[Y (1)(1) || T
T =
= 0]
0]
P (T == 1) E[YE[Y (0) || T
T = 1]
1] P (T(T = 0)E[Y (0)
|T =
0]
P
P (T
(T = 1)
1) E[Y (0)(0) | T == 1] PP (T == 0)0) E[YE[Y (0)(0) || T
T =
= 0]
0]
=P
P (T
(T =
= 1)
1) E[YE[Y || T
T =
= 1]
1] +
+P
P (T
(T =
= 0)
0) E[YE[Y (1)
|| T
=
0]
=
(1)
T
=
0]
Observational = P (T = 1) E[Y | T = 1] + P (T = 0) E[Y (1) | T = 0]
P (T
(T =
= 1)
1) E[YE[Y (0)(0) || T
T =
= 1]
1] P
P (T
(T =
= 0)
E[Y
|| T
=
0]
P
0)
E[Y
T
=
P (T = 1) E[Y (0) | T = 1] P (T = 0) E[Y | T = 0]0]

Brady Neal

Bounds: No-Assumptions Bound

8 / 48Observational-Counterfactual Decomposition

Proof:<latexit 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sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit><latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit><latexit sha1_base64="6gRplhWUyi4eKATeimkwFnu3lY8=">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</latexit>

E[Y
(1)
E[Y
E[Y (1)(1)= E[Y (1)](1)] E[Y (0)](0)]
Y (0)] =Y
Y (0)](0)] = E[YE[Y (1)] E[YE[Y (0)]= P (T(T = 1)
1) E[Y (1)(1) | T == 1] ++ P (T(T = 0)0) E[Y (1)
|T =
0]
=
=P
P (T == 1) E[YE[Y (1) || T
T = 1]
1] + PP (T == 0) E[YE[Y (1)(1) || T
T =
= 0]
0]
P (T == 1) E[YE[Y (0) || T
T = 1]
1] P (T(T = 0)E[Y (0)
|T =
0]
P
P (T
(T = 1)
1) E[Y (0)(0) | T == 1] PP (T == 0)0) E[YE[Y (0)(0) || T
T =
= 0]
0]
=P
P (T
(T =
= 1)
1) E[YE[Y || T
T =
= 1]
1] +
+P
P (T
(T =
= 0)
0) E[YE[Y (1)
|| T
=
0]
=
(1)
T
=
0]
Observational = P (T = 1) E[Y | T = 1] + P (T = 0) E[Y (1) | T = 0]
P (T
(T =
= 1)
1) E[YE[Y (0)(0) || T
T =
= 1]
1] P
P (T
(T =
= 0)
E[Y
|| T
=
0]
P
0)
E[Y
T
=
P (T = 1) E[Y (0) | T = 1] P (T = 0) E[Y | T = 0]0]Counterfactual

Brady Neal

Bounds: No-Assumptions Bound

8 / 48Observational-Counterfactual Decomposition<latexit sha1_base64="UDEveKn4XqnaZ1Ch83og9MxZEYE=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0](1

where ⇡ , P (T = 1)
<latexit sha1_base64="groc7N+KaMxnNILgpGbGW0JnGbw=">AAAT3XicrVhLb9tGEN6kr8R9OdGxFyZKgRSgFclN4VwEBLATFEUDOIVfbRQYFLWSCPEVkrLqCDoWvRS99lf02v6W/pvOfLsUST1IKYgEiavZmW9m57W76oauEyfN5n83bn7w4Ucff3Lr9s6nn33+xZe7d+6excE4suWpHbhBdNG1Yuk6vjxNnMSVF2EkLa/ryvPu6JDnz69kFDuBf5Jch/K1Zw18p+/YVkKky917ndAxOknkWP7AlW+MTviQfkkr8aSfGG2j9c3lbr3ZaOJlLA9aelAX+nUc3Ln9k+iIngiELcbCE1L4IqGxKywR0/uVaImmCIn2WkyJFtHIwbwUM7FDsmPiksRhEXVE3wP69UpTffrNmDGkbdLi0iciSUN8rXl6NO6Dqp6s38jxrtMxBTbbeE3Prsb0iJqIIVGr5FLOzeW69PYqVp3QGp5gtQ6tJASF/WAX1tynp0u/E1ohf18Tp6RRj6QiGtlEc4mqKKwjoqfyPPtmiEhY4JM0YqvL7C6zmucDeo8Iy6JxDEvZVkM813HxoVnCVuZxEdP1iL/SCpV9ZVj9+RocxF2tgnlPiDISb2lURC7Tmc+u9VyJRp6t0MX8CTh84o6JEiDzHfKfQxzFXLUJk3VYiOQAawmRQQ2N/APm02wOKZ57sCdG1AzkmwM9oa6szO7UTpeeXWBHJD+luSH0sC9MskcCO0L2pl40wR3Rrwl+2bDRXqA3UJkcT5PWxNllEm5AM84cS3lC+TO1SFk3JZqh33viR+iXFHcTXuG8Nkk2gIUqi2Po8nXs29RX2C8efTOVPZtSTOjwYMcQ2TIiWqZPIbG+R7SKGPXjatQpPV14LQSKCe3snwnGHtbEER1D94TGPehhaYN6XUMcaDtmBa3sGwfdaVmrgfzj2lWdcVEry/moVENnZRuamuKxXjFHgO3Oe83W3VD15Birmy1Y1YX9WdfgHC3ax1JZfIqx6iEbh5Aq4rIXV69z1Vr2UScmYsm+HxBHG1J9TY9Rm6HWtJOrv0NdJQHiYePp6ZxQ9c7aJgszU6J72lMu8ak65z2JZ94QpoWc6sB7+fkyxGdaPt0NuONPQS2XO0NlFWUjGk3nM+XyDkYS/lEjjrjCC+HrEGu5N/8YuZlNsd8Pnr+EyBmbaKRN0XvIjdUrPxF16gV1yqqiPznSTIlwOglp/GDO+YBwmXcELT5RuMKn8/lZZQyOiDLTHutjLq3DzAYHna8Hzo7OPMWhKipZ4OCqLtfKebkqbxS9XLY3z/eirKKXyzLlCrnp6H4QA+uiQi5BHFjCy2XpSYVUgD1B0ZXNP29gn4+9dqzzItIWnldIeugxalWBrsoXlfYl6IKR9gnL/PIOHrxC/5vhBLStHzPZZCtvZnLX7+TTTH6ypWczSW9L/2aSbyskJfYGZ05jmZeVNcVcx5VcKp+q9gRJu0l6flHVZuK7S17kPbunO5Oh9xvOCrU7ttEDzI368/K+w5XcqqzjMbpvF30oguZeSe2cvje0LIJjvZenu3l6Co9zfd6BxuyEPsVpaqJ3hPRctQdPKl98Tz3lBZ1VeHwovqVzUou+nxd2j01R+b4y1n0nj7tPyAfU94+wP2yPG+pbxV7h7lHU8Yw0pJ+ZvqVl3K4+S69D2kbvamQVvd7SmX5Rkzq1DYlL3WSuKxCzfFiNVx7f1cjqH4cAJ+nsPvM+kPN3tc3wspxZf7Ouuu+vv4WG9OzTt4vOGGveI2Sdiw4pqVbVTVmdQg6X7vycv/tUFZy70/t3F/QV8RdvkQPcAsfEkfGo20ACaUXLI667j4aQSfSpK8bdLbuhNgo3ToW3aFucyytP47V1h5K66+5c7tZbi/9iLQ/O9hut7xrNl4/rT5/of7huia/EffGQ/HQgnlJlHpNfbfG7+Fv8I/6tXdZ+q/1R+1Ox3ryhZWqi8Kr99T/AGdsl</latexit>⇡) E[Y | T = 0]

Proof:<latexit 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E[Y
(1)
E[Y
E[Y (1)(1)= E[Y (1)](1)] E[Y (0)](0)]
Y (0)] =Y
Y (0)](0)] = E[YE[Y (1)] E[YE[Y (0)]= P (T(T = 1)
1) E[Y (1)(1) | T == 1] ++ P (T(T = 0)0) E[Y (1)
|T =
0]
=
=P
P (T == 1) E[YE[Y (1) || T
T = 1]
1] + PP (T == 0) E[YE[Y (1)(1) || T
T =
= 0]
0]
P (T == 1) E[YE[Y (0) || T
T = 1]
1] P (T(T = 0)E[Y (0)
|T =
0]
P
P (T
(T = 1)
1) E[Y (0)(0) | T == 1] PP (T == 0)0) E[YE[Y (0)(0) || T
T =
= 0]
0]
=P
P (T
(T =
= 1)
1) E[YE[Y || T
T =
= 1]
1] +
+P
P (T
(T =
= 0)
0) E[YE[Y (1)
|| T
=
0]
=
(1)
T
=
0]
Observational = P (T = 1) E[Y | T = 1] + P (T = 0) E[Y (1) | T = 0]
P (T
(T =
= 1)
1) E[YE[Y (0)(0) || T
T =
= 1]
1] P
P (T
(T =
= 0)
E[Y
|| T
=
0]
P
0)
E[Y
T
=
P (T = 1) E[Y (0) | T = 1] P (T = 0) E[Y | T = 0]0]Counterfactual

Brady Neal

Bounds: No-Assumptions Bound

8 / 48

No-Assumptions Bound
Observational-Counterfactual Decomposition<latexit sha1_base64="UDEveKn4XqnaZ1Ch83og9MxZEYE=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]
where ⇡ , P (T = 1)⇡) E[Y (1) | T = 0](1<latexit sha1_base64="groc7N+KaMxnNILgpGbGW0JnGbw=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound⇡) E[Y | T = 0]

9 / 48

No-Assumptions Bound
Observational-Counterfactual Decomposition<latexit sha1_base64="UDEveKn4XqnaZ1Ch83og9MxZEYE=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]
where ⇡ , P (T = 1)⇡) E[Y (1) | T = 0](1<latexit sha1_base64="groc7N+KaMxnNILgpGbGW0JnGbw=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound⇡) E[Y | T = 0]

9 / 48

No-Assumptions Bound
Observational-Counterfactual Decomposition<latexit sha1_base64="UDEveKn4XqnaZ1Ch83og9MxZEYE=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]
where ⇡ , P (T = 1)⇡) E[Y (1) | T = 0](1⇡) E[Y | T = 0]<latexit sha1_base64="groc7N+KaMxnNILgpGbGW0JnGbw=">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</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit sha1_base64="BTumfeG5h10ehdAgjCXnSCWV/KM=">AAAUUnicrVjbbttGEF0lvSRu2jix3/rCWilgI7QquSmcFwEBbAdF0QBO4VtrGYZEUhYh3kJSVh1Bv9Ev60t/pehDZ84uRVIXUgosQeJqdubM7Nx2V53AsaO4Xv+n8uDhZ59/8eWjx2tfPfn6m6frz56fRf4gNKxTw3f88KLTjizH9qzT2I4d6yIIrbbbcazzTv+A589vrTCyfe8kvgusK7d949ld22jHRLpe/6t1dNnyB7Hhu9Z2Y0fb1Sa/6jtXWsuxPmitwNZaupbh1FqubWqtmDTFruXFWlNrXGkvte0GAwT2DvN35JiHbRrm5oqx6lfX69V6rY6XNjtoqEFVqNex/+zxb6IlTOELQwyEKyzhiZjGjmiLiN6XoiHqIiDalRgRLaSRjXlLjMUayQ6IyyKONlH79H1Dvy4V1aPfjBlB2iAtDn1CktTE94rHpHEXVPlk/VqGd5GOEbDZxjt6dhSmS9RY9IhaJpdwLi/XobdbsuqY1vAaq7VpJQEo7Acjt+YuPR36HdMK+fuOOC0amSQV0sggmkNUSWEdIT2l59k3PUSiDT6LRmx1kd1FVvO8T+8+YbVpHMFStlUTb1VcPGi2YCvzOIjpYsQ/aYXSviKs7mQNNuIuV8G8J0Tpi480yiMX6cxm12KuWCGP5+hi/hgcHnFHRPGR+Tb5zyaOfK4ahMk62ojkDdYSIINqCvkXzCfZHFA8d2FPhKhpyDcbegJVWandiZ0OPTvADkl+RHM96GFf6GSPBewQ2Zt4UQd3SL+G+GXARmOKXkNlcjx1WhNnl064Ps3YEyzpCenPxCJp3Yhomnrvil+h36K46/AK57VOsj4slFkcQZenYt+kvsJ+cembqezZhKJDhws7esiWPtFSfRKJ9f1Aq4hQP45CHdHTgdcCoOjQzv4ZYuxiTRzRAXQPaWxCD0tr1OtqYl/ZMc5pZd/Y6E6zWjXkH9eu7IzTWlnOQ6VqKiub0FQXr9SKOQJsd9ZrhuqGsidHWN14yqoO7E+7Budo3j6WSuOTj5WJbOxBKo/LXpy/znlr2UOd6Igl+/6GOJqQ6ip6hNoMlKa1TP0dqCrxEQ8DT1flhKx31jacmhkR3VWecohP1jnvSTzzgTDbyKkWvJedL0I8UvLJbsAdfwRqsdwZKisvG9JoNJkplrcxsuAfOeKIS7wAvg6wlu8mHy0zsyz2/eB5M4icsbFCWhbdRG7MX/mJqFIvqFJW5f3JkWZKiNNJQOMXE84XhMu8fWjxiMIVPprMj0tjcEiUsfJYF3NJHaY22Oh8JjhbKvMkh6yoeIqDq7pYK+flvLyR9GJZc5LveVlJL5Zlyi1y01b9IALWRYlcjDiwhJvJ0pMSKR97gqRLm39fwj4Pe+1A5UWoLDwvkXTRY+SqfFWV70rti9EFQ+UTlvnjEzx4i/43xgloVT+msvFK3kzl7j7Jp6n8cEXPppLuiv5NJT+WSFrYG+wJjWXel9YUcx2Xcsl8KtsTLNpNkvOLrDYd3x3yIu/ZpupMmtpvOCvk7thED9CX6s+z+w5XcqO0jgfovh30oRCazYLaOb03tDSCA7WXJ7t5cgqPMn3ehsb0hD7CaWqodoTkXLULT0pf/Ew95R2dVXh8IH6kc1KDvt/mdo9lUfm+MlB9J4u7R8j71PcPsT+sjhuoW8Vu7u6R13FEGpLPWN3SUm5HnaUXIa2idz6yjJ45c6af1iRPbT3ikjeZuxLENB/m4xXHdz6y/MfBx0k6vc/cB3L2rrYcXpozi2/WZff9xbfQgJ5d+nbQGSPFe4isc9AhLapVeVOWp5CDmTs/5+8eVQXn7mjr+ZS+PP70LfIGt8ABcaQ88jYQQ1rSsoiL7qMBZGJ16opwd0tvqLXcjVPiTdsWZfLKVXhN1aEs1XXXrterjel/sWYHZ3u1xk+1+vtX1Tev1T9cj8S3Yktsk5/2xRuqzGPyqyH+q2xVXlb0jb83/t2sbD6UrA8qSmZD5F6bT/4Hsvz8JA==</latexit><latexit sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">AAAUUnicrVjbbttGEF0lvSRu2jix3/rCWilgI7QquSmcFwEBbAdF0QBO4VtrGYZEUhIh3kJSVh1Bv9Ev60t/pehDZ84uRVIXUg4sQeJqdubM7Nx2V53AsaO4Xv+n8uDhZ59/8eWjx2tfPfn6m6frz56fRf4wNKxTw3f88KLTjizH9qzT2I4d6yIIrbbbcazzzuCA589vrDCyfe8kvg2sK7fd8+yubbRjIl2v/9U6umz5w9jwXWu7saPtatNf9Z0rrdWzPmitwNZaupbh1FqubWqtmDTFruXFWlNrXGkvte0GAwT2DvO35ZiHHRrm5oqx6lfX69V6rY6XNj9oqEFVqNex/+zxb6IlTOELQwyFKyzhiZjGjmiLiN6XoiHqIiDalRgTLaSRjXlLTMQayQ6JyyKONlEH9N2jX5eK6tFvxowgbZAWhz4hSWrie8Vj0rgLqnyyfi3Du0zHGNhs4y09OwrTJWos+kQtk0s4V5fr0NstWXVMa3iN1dq0kgAU9oORW3OXng79jmmF/H1LnBaNTJIKaWQQzSGqpLCOkJ7S8+ybPiLRBp9FI7a6yO4iq3nep/eAsNo0jmAp26qJtyouHjRbsJV5HMR0OeKftEJpXxFWd7oGG3GXq2DeE6IMxEca5ZGLdGazazlXrJAnC3QxfwwOj7gjovjIfJv8ZxNHPlcNwmQdbUSyh7UEyKCaQv4F80k2BxTPXdgTIWoa8s2GnkBVVmp3YqdDzw6wQ5If01wfetgXOtljATtE9iZe1MEd0q8Rfhmw0Zih11CZHE+d1sTZpROuTzP2FEt6QvozsUhaNyaapt674lfotyjuOrzCea2TrA8LZRZH0OWp2Depr7BfXPpmKns2oejQ4cKOPrJlQLRUn0RifT/QKiLUj6NQx/R04LUAKDq0s39GGLtYE0d0CN0jGpvQw9Ia9bqa2Fd2THJa2Tc2utO8Vg35x7UrO+OsVpbzUKmaysomNNXFK7VijgDbnfWaobqh7MkRVjeZsaoD+9OuwTmat4+l0vjkY2UiG/uQyuOyFxevc9Fa9lAnOmLJvu8RRxNSXUWPUJuB0rSWqb8DVSU+4mHg6aqckPXO2kYzM2Oiu8pTDvHJOuc9iWc+EGYbOdWC97LzRYhHSj7ZDbjjj0EtljtDZeVlQxqNpzPF8jZGFvwjRxxxiRfA1wHW8t30o2VmVsW+HzxvDpEzNlZIq6KbyI3FKz8RVeoFVcqqvD850kwJcToJaPxiyvmCcJl3AC0eUbjCx9P5SWkMDokyUR7rYi6pw9QGG53PBGdLZZ7kkBUVz3BwVRdr5bxclDeSXixrTvM9LyvpxbJMuUFu2qofRMC6KJGLEQeWcDNZelIi5WNPkHRp8+8r2Odhrx2qvAiVheclki56jFyVr6ryXal9MbpgqHzCMn98ggdv0P8mOAHd1Y+pbHwnb6Zyt5/k01R+dEfPppLuHf2bSn4skbSwN9hTGsu8L60p5jou5ZL5VLYnWLSbJOcXWW06vjvkRd6zTdWZNLXfcFbI3bGJHqCv1J/n9x2u5EZpHQ/RfTvoQyE0mwW1c3pvaGkEh2ovT3bz5BQeZfq8DY3pCX2M09RI7QjJuWoXnpS++Jl6yjs6q/D4QPxI56QGfb/N7R6rovJ9Zaj6ThZ3j5D3qe8fYn+4O26gbhW7ubtHXscRaUg+E3VLS7kddZZehnQXvYuRZfTMuTP9rCZ5ausTl7zJ3JYgpvmwGK84vouR5T8OPk7S6X3mPpCzd7XV8NKcWX6zLrvvL7+FBvTs0reDzhgp3kNknYMOaVGtypuyPIUczN35OX/3qCo4d8dbz2f05fFnb5E93AKHxJHyyNtADGlJyyIuu48GkInVqSvC3S29odZyN06JN2tblMkrV+E1VYeyVNddu16vNmb/xZofnO3VGj/V6u9fVd+8Vv9wPRLfii2xTX7aF2+oMo/Jr4b4r7JVeVnRN/7e+HezsvlQsj6oKJkNkXttPvkfT/D8Hw==</latexit>

Brady Neal

Bounds: No-Assumptions Bound

9 / 48

No-Assumptions Bound
Observational-Counterfactual Decomposition<latexit sha1_base64="UDEveKn4XqnaZ1Ch83og9MxZEYE=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]
where ⇡ , P (T = 1)⇡) E[Y (1) | T = 0](1⇡) E[Y | T = 0]<latexit sha1_base64="groc7N+KaMxnNILgpGbGW0JnGbw=">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</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit sha1_base64="BTumfeG5h10ehdAgjCXnSCWV/KM=">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</latexit><latexit sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

9 / 48

No-Assumptions Bound
Observational-Counterfactual Decomposition<latexit sha1_base64="UDEveKn4XqnaZ1Ch83og9MxZEYE=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]
where ⇡ , P (T = 1)⇡) E[Y (1) | T = 0](1⇡) E[Y | T = 0]<latexit sha1_base64="groc7N+KaMxnNILgpGbGW0JnGbw=">AAAT3XicrVhLb9tGEN6kr8R9OdGxFyZKgRSgFclN4VwEBLATFEUDOIVfbRQYFLWSCPEVkrLqCDoWvRS99lf02v6W/pvOfLsUST1IKYgEiavZmW9m57W76oauEyfN5n83bn7w4Ucff3Lr9s6nn33+xZe7d+6excE4suWpHbhBdNG1Yuk6vjxNnMSVF2EkLa/ryvPu6JDnz69kFDuBf5Jch/K1Zw18p+/YVkKky917ndAxOknkWP7AlW+MTviQfkkr8aSfGG2j9c3lbr3ZaOJlLA9aelAX+nUc3Ln9k+iIngiELcbCE1L4IqGxKywR0/uVaImmCIn2WkyJFtHIwbwUM7FDsmPiksRhEXVE3wP69UpTffrNmDGkbdLi0iciSUN8rXl6NO6Dqp6s38jxrtMxBTbbeE3Prsb0iJqIIVGr5FLOzeW69PYqVp3QGp5gtQ6tJASF/WAX1tynp0u/E1ohf18Tp6RRj6QiGtlEc4mqKKwjoqfyPPtmiEhY4JM0YqvL7C6zmucDeo8Iy6JxDEvZVkM813HxoVnCVuZxEdP1iL/SCpV9ZVj9+RocxF2tgnlPiDISb2lURC7Tmc+u9VyJRp6t0MX8CTh84o6JEiDzHfKfQxzFXLUJk3VYiOQAawmRQQ2N/APm02wOKZ57sCdG1AzkmwM9oa6szO7UTpeeXWBHJD+luSH0sC9MskcCO0L2pl40wR3Rrwl+2bDRXqA3UJkcT5PWxNllEm5AM84cS3lC+TO1SFk3JZqh33viR+iXFHcTXuG8Nkk2gIUqi2Po8nXs29RX2C8efTOVPZtSTOjwYMcQ2TIiWqZPIbG+R7SKGPXjatQpPV14LQSKCe3snwnGHtbEER1D94TGPehhaYN6XUMcaDtmBa3sGwfdaVmrgfzj2lWdcVEry/moVENnZRuamuKxXjFHgO3Oe83W3VD15Birmy1Y1YX9WdfgHC3ax1JZfIqx6iEbh5Aq4rIXV69z1Vr2UScmYsm+HxBHG1J9TY9Rm6HWtJOrv0NdJQHiYePp6ZxQ9c7aJgszU6J72lMu8ak65z2JZ94QpoWc6sB7+fkyxGdaPt0NuONPQS2XO0NlFWUjGk3nM+XyDkYS/lEjjrjCC+HrEGu5N/8YuZlNsd8Pnr+EyBmbaKRN0XvIjdUrPxF16gV1yqqiPznSTIlwOglp/GDO+YBwmXcELT5RuMKn8/lZZQyOiDLTHutjLq3DzAYHna8Hzo7OPMWhKipZ4OCqLtfKebkqbxS9XLY3z/eirKKXyzLlCrnp6H4QA+uiQi5BHFjCy2XpSYVUgD1B0ZXNP29gn4+9dqzzItIWnldIeugxalWBrsoXlfYl6IKR9gnL/PIOHrxC/5vhBLStHzPZZCtvZnLX7+TTTH6ypWczSW9L/2aSbyskJfYGZ05jmZeVNcVcx5VcKp+q9gRJu0l6flHVZuK7S17kPbunO5Oh9xvOCrU7ttEDzI368/K+w5XcqqzjMbpvF30oguZeSe2cvje0LIJjvZenu3l6Co9zfd6BxuyEPsVpaqJ3hPRctQdPKl98Tz3lBZ1VeHwovqVzUou+nxd2j01R+b4y1n0nj7tPyAfU94+wP2yPG+pbxV7h7lHU8Yw0pJ+ZvqVl3K4+S69D2kbvamQVvd7SmX5Rkzq1DYlL3WSuKxCzfFiNVx7f1cjqH4cAJ+nsPvM+kPN3tc3wspxZf7Ouuu+vv4WG9OzTt4vOGGveI2Sdiw4pqVbVTVmdQg6X7vycv/tUFZy70/t3F/QV8RdvkQPcAsfEkfGo20ACaUXLI667j4aQSfSpK8bdLbuhNgo3ToW3aFucyytP47V1h5K66+5c7tZbi/9iLQ/O9hut7xrNl4/rT5/of7huia/EffGQ/HQgnlJlHpNfbfG7+Fv8I/6tXdZ+q/1R+1Ox3ryhZWqi8Kr99T/AGdsl</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

9 / 48

No-Assumptions Bound
Observational-Counterfactual Decomposition<latexit sha1_base64="UDEveKn4XqnaZ1Ch83og9MxZEYE=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]
where ⇡ , P (T = 1)⇡) E[Y (1) | T = 0](1⇡) E[Y | T = 0]<latexit sha1_base64="groc7N+KaMxnNILgpGbGW0JnGbw=">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</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

9 / 48

No-Assumptions Bound
Observational-Counterfactual Decomposition<latexit sha1_base64="UDEveKn4XqnaZ1Ch83og9MxZEYE=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]
where ⇡ , P (T = 1)⇡) E[Y (1) | T = 0](1⇡) E[Y | T = 0]<latexit sha1_base64="groc7N+KaMxnNILgpGbGW0JnGbw=">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</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

9 / 48

No-Assumptions Bound
Observational-Counterfactual Decomposition<latexit sha1_base64="UDEveKn4XqnaZ1Ch83og9MxZEYE=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]
where ⇡ , P (T = 1)⇡) E[Y (1) | T = 0](1⇡) E[Y | T = 0]<latexit sha1_base64="groc7N+KaMxnNILgpGbGW0JnGbw=">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</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

9 / 48

No-Assumptions Bound
Observational-Counterfactual Decomposition<latexit sha1_base64="UDEveKn4XqnaZ1Ch83og9MxZEYE=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]
where ⇡ , P (T = 1)⇡) E[Y (1) | T = 0](1⇡) E[Y | T = 0]<latexit sha1_base64="groc7N+KaMxnNILgpGbGW0JnGbw=">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</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

9 / 48

No-Assumptions Bound
Observational-Counterfactual Decomposition<latexit sha1_base64="UDEveKn4XqnaZ1Ch83og9MxZEYE=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]
where ⇡ , P (T = 1)⇡) E[Y (1) | T = 0](1⇡) E[Y | T = 0]<latexit sha1_base64="groc7N+KaMxnNILgpGbGW0JnGbw=">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</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

9 / 48

No-Assumptions Bound
Observational-Counterfactual Decomposition<latexit sha1_base64="UDEveKn4XqnaZ1Ch83og9MxZEYE=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]
where ⇡ , P (T = 1)⇡) E[Y (1) | T = 0](1⇡) E[Y | T = 0]<latexit sha1_base64="groc7N+KaMxnNILgpGbGW0JnGbw=">AAAT3XicrVhLb9tGEN6kr8R9OdGxFyZKgRSgFclN4VwEBLATFEUDOIVfbRQYFLWSCPEVkrLqCDoWvRS99lf02v6W/pvOfLsUST1IKYgEiavZmW9m57W76oauEyfN5n83bn7w4Ucff3Lr9s6nn33+xZe7d+6excE4suWpHbhBdNG1Yuk6vjxNnMSVF2EkLa/ryvPu6JDnz69kFDuBf5Jch/K1Zw18p+/YVkKky917ndAxOknkWP7AlW+MTviQfkkr8aSfGG2j9c3lbr3ZaOJlLA9aelAX+nUc3Ln9k+iIngiELcbCE1L4IqGxKywR0/uVaImmCIn2WkyJFtHIwbwUM7FDsmPiksRhEXVE3wP69UpTffrNmDGkbdLi0iciSUN8rXl6NO6Dqp6s38jxrtMxBTbbeE3Prsb0iJqIIVGr5FLOzeW69PYqVp3QGp5gtQ6tJASF/WAX1tynp0u/E1ohf18Tp6RRj6QiGtlEc4mqKKwjoqfyPPtmiEhY4JM0YqvL7C6zmucDeo8Iy6JxDEvZVkM813HxoVnCVuZxEdP1iL/SCpV9ZVj9+RocxF2tgnlPiDISb2lURC7Tmc+u9VyJRp6t0MX8CTh84o6JEiDzHfKfQxzFXLUJk3VYiOQAawmRQQ2N/APm02wOKZ57sCdG1AzkmwM9oa6szO7UTpeeXWBHJD+luSH0sC9MskcCO0L2pl40wR3Rrwl+2bDRXqA3UJkcT5PWxNllEm5AM84cS3lC+TO1SFk3JZqh33viR+iXFHcTXuG8Nkk2gIUqi2Po8nXs29RX2C8efTOVPZtSTOjwYMcQ2TIiWqZPIbG+R7SKGPXjatQpPV14LQSKCe3snwnGHtbEER1D94TGPehhaYN6XUMcaDtmBa3sGwfdaVmrgfzj2lWdcVEry/moVENnZRuamuKxXjFHgO3Oe83W3VD15Birmy1Y1YX9WdfgHC3ax1JZfIqx6iEbh5Aq4rIXV69z1Vr2UScmYsm+HxBHG1J9TY9Rm6HWtJOrv0NdJQHiYePp6ZxQ9c7aJgszU6J72lMu8ak65z2JZ94QpoWc6sB7+fkyxGdaPt0NuONPQS2XO0NlFWUjGk3nM+XyDkYS/lEjjrjCC+HrEGu5N/8YuZlNsd8Pnr+EyBmbaKRN0XvIjdUrPxF16gV1yqqiPznSTIlwOglp/GDO+YBwmXcELT5RuMKn8/lZZQyOiDLTHutjLq3DzAYHna8Hzo7OPMWhKipZ4OCqLtfKebkqbxS9XLY3z/eirKKXyzLlCrnp6H4QA+uiQi5BHFjCy2XpSYVUgD1B0ZXNP29gn4+9dqzzItIWnldIeugxalWBrsoXlfYl6IKR9gnL/PIOHrxC/5vhBLStHzPZZCtvZnLX7+TTTH6ypWczSW9L/2aSbyskJfYGZ05jmZeVNcVcx5VcKp+q9gRJu0l6flHVZuK7S17kPbunO5Oh9xvOCrU7ttEDzI368/K+w5XcqqzjMbpvF30oguZeSe2cvje0LIJjvZenu3l6Co9zfd6BxuyEPsVpaqJ3hPRctQdPKl98Tz3lBZ1VeHwovqVzUou+nxd2j01R+b4y1n0nj7tPyAfU94+wP2yPG+pbxV7h7lHU8Yw0pJ+ZvqVl3K4+S69D2kbvamQVvd7SmX5Rkzq1DYlL3WSuKxCzfFiNVx7f1cjqH4cAJ+nsPvM+kPN3tc3wspxZf7Ouuu+vv4WG9OzTt4vOGGveI2Sdiw4pqVbVTVmdQg6X7vycv/tUFZy70/t3F/QV8RdvkQPcAsfEkfGo20ACaUXLI667j4aQSfSpK8bdLbuhNgo3ToW3aFucyytP47V1h5K66+5c7tZbi/9iLQ/O9hut7xrNl4/rT5/of7huia/EffGQ/HQgnlJlHpNfbfG7+Fv8I/6tXdZ+q/1R+1Ox3ryhZWqi8Kr99T/AGdsl</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

9 / 48

No-Assumptions Bound
Observational-Counterfactual Decomposition<latexit sha1_base64="UDEveKn4XqnaZ1Ch83og9MxZEYE=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]
where ⇡ , P (T = 1)⇡) E[Y (1) | T = 0](1⇡) E[Y | T = 0]<latexit sha1_base64="groc7N+KaMxnNILgpGbGW0JnGbw=">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</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

9 / 48

No-Assumptions Bound
Observational-Counterfactual Decomposition<latexit sha1_base64="UDEveKn4XqnaZ1Ch83og9MxZEYE=">AAAUrHicrVhZb9tGEF6lV+JeTuq3vmzjNHBQSpXcFM6LgAC2g6JoAKfw1ZiCIZGUTZhXSMqqo+qx/Y/9F/0Jnfl2KZK6KAeWIHE1O/PN7Fy7q17kuUnabP5bu/fRx598+tn9B2uff/HlV1+vP3x0nISD2HKOrNAL49NeN3E8N3COUjf1nNModrp+z3NOele7PH9y7cSJGwaH6U3kdPzuReD2XaubEul8/T9z/8wMB6kV+s5W65msy8mv5rOOfNqWZuRK05AFPmn6ri3NlPSkvhOksi1bHfmD3GqxeOQ+m+Jn3FmRZkeaQRgM/J4TS9Nck/R6ar4bdG2FMg3SnAdCeusL9c5Ver6+2Ww08ZKzg5YebAr9OggfPvhdmMIWobDEQPjCEYFIaeyJrkjofSZaoikionXEiGgxjVzMO2Is1kh2QFwOcXSJekXfF/TrTFMD+s2YCaQt0uLRJyZJKb7XPDaN+6CqJ+uXBd5FOkbAZhtv6NnTmD5RU3FJ1Cq5jHN1uR69/YpVp7SGF1itSyuJQGE/WKU19+np0e+UVsjfN8Tp0MgmqZhGFtE8oioK64jpqTzPvrlEJLrgc2jEVi+ze5nVPB/S+4qwujROYCnbKsUrHZcAmh3YyjweYroY8U9aobJvGVZ/sgYXcVerYN5DolyJ9zQqIy/TWcyuxVypRh7P0cX8KTgC4k6IEiLzXfKfSxzlXLUIk3V0EckLrCVCBjU08q+Yz7I5onjWYU+CqEnkmws9ka6s3O7MTo+ePWDHJD+iuUvoYV8YZI8D7BjZm3nRAHdMv4b4ZcFGa4reQGVyPA1aE2eXQbghzbgTLOUJ5c/MImXdiGhSv+viN+h3KO4GvMJ5bZBsCAtVFifQFejYt6mvsF98+mYqezajGNDhw45LZMsV0XJ9Con1/UirSFA/nkYd0dOD1yKgGNDO/hli7GNNHNEBdA9pbEMPS0vqdQ2xo+0Yl7Syb1x0p1mtEvnHtas647RWlgtQqVJnZRuamuK5XjFHgO0ues3S3VD15ASrG09Z1YP9edfgHC3bx1J5fMqxspGNl5Aq47IX569z3lq2UScGYsm+vyCONqT6mp6gNiOtaa1Qf7u6SkLEw8LT1zmh6p21DadmRkT3tac84lN1znsSz7wjzC5yyoT3ivPLEPe1fLYbcMcfgbpc7hiVVZaNaTSazCyXdzFy4B814ogrvAi+jrCW7yYfWZhZFftu8IIZRM7YVCOtim4jN+av/FBsUi/YpKwq+5MjzZQYp5OIxk8mnE8Il3mvoCUgClf4aDI/rozBHlHG2mN9zGV1mNvgovPZ4DR15ikOVVHpFAdX9XKtnJfz8kbRl8vak3wvyyr6clmmXCM3Xd0PEmCdVsiliANL+IUsPayQCrEnKLqy+Y8V7Auw1w50XsTawpMKSR89Rq0q1FX5utK+FF0w1j5hmbcf4MFr9L8xTkC39WMum97Km7nczQf5NJcf3tKzuaR/S//mku8rJB3sDe6ExjJvKmuKuQ4quVQ+Ve0JDu0m2flFVZuB7x55kfdsW3cmqfcbzgq1O7bRA4yV+vPsvsOV3Kqs4wG6bw99KIZme0ntHN0ZWh7Bgd7Ls908O4UnhT7vQmN+Qh/hNDXUO0J2rqrDk8oXv1BPeU1nFR7vip/onNSi71el3WNVVL6vDHTfKeJuE/IO9f097A+3x430raJeunuUdeyThuwz1re0nNvTZ+lFSLfROx9ZRc+eOdNPa1KntkviUjeZmwrEPB/m4y2P73xk9Y9DiJN0fp+5C+TiXW01vDxnFt+sq+77i2+hET379O2hMyaadw9Z56FDOlSr6qasTiG7M3d+zt9tqgrO3dHjR1P6yvjTt8gL3AIHxJHzqNtACmlFKyIuuo9GkEn1qSvB3S2/oTZKN06FN21bUsgrX+O1dYdydNddO1/fbE3/izU7ON5utH5uNN8833z5Qv/DdV98Kx6LLfLTjnhJlXlAfrVqx7W/an/X/tlobBxunG10FOu9mpb5RpReG/3/AUyFGfk=</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]
where ⇡ , P (T = 1)⇡) E[Y (1) | T = 0](1⇡) E[Y | T = 0]<latexit sha1_base64="groc7N+KaMxnNILgpGbGW0JnGbw=">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</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit><latexit sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

9 / 48

No-Assumptions Interval Length

Brady Neal

Bounds: No-Assumptions Bound

10 / 48

No-Assumptions Interval Length
Recall trivial bound:<latexit sha1_base64="yhX9WtXCdiGLM3RE0rncm1vlvfw=">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</latexit>ab  E[Y (1)Y (0)]  baTrivial length limit: 2(ba)<latexit sha1_base64="xDUWc/MYNypUpJO9sx67/fWnYkY=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

10 / 48

No-Assumptions Interval Length
Recall trivial bound:<latexit sha1_base64="yhX9WtXCdiGLM3RE0rncm1vlvfw=">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</latexit>ab  E[Y (1)Y (0)]  baTrivial length limit: 2(ba)<latexit sha1_base64="xDUWc/MYNypUpJO9sx67/fWnYkY=">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</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit><latexit sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

10 / 48

No-Assumptions Interval Length
Recall trivial bound:<latexit sha1_base64="yhX9WtXCdiGLM3RE0rncm1vlvfw=">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</latexit>ab  E[Y (1)Y (0)]  baTrivial length limit: 2(ba)<latexit sha1_base64="xDUWc/MYNypUpJO9sx67/fWnYkY=">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</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

No-assumptions interval length: (1<latexit sha1_base64="n+16VmBiHMgfmUjdepzMWj9en4Q=">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</latexit>

Brady Neal⇡) b + ⇡ b⇡aBounds: No-Assumptions Bound(1⇡) a = ba
10 / 48

No-Assumptions Interval Length
Recall trivial bound:<latexit sha1_base64="yhX9WtXCdiGLM3RE0rncm1vlvfw=">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</latexit>ab  E[Y (1)Y (0)]  baTrivial length limit: 2(ba)<latexit sha1_base64="xDUWc/MYNypUpJO9sx67/fWnYkY=">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</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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No-assumptions interval length: (1<latexit sha1_base64="n+16VmBiHMgfmUjdepzMWj9en4Q=">AAAT+XicrVhLb9tGEN6kr8R9OdGxFzZKgRSlVMlN4VwEBLATFEUDOIVfbRQYJEVZhPgqSVl1CP2Yopei1/6KXttT/01nvl2KpB6kFFiCxNXszDez89pdmaHrxEmn89+t2++8+977H9y5u/PhRx9/8unuvfuncTCJLPvECtwgOjeN2HYd3z5JnMS1z8PINjzTtc/M8QHPn13ZUewE/nFyHdqvPePSd4aOZSREutjtPepqLa0fOl9qfV0zta94LIetbGjQsMRmaD3MGxe7zU67g5e2POiqQVOo11Fw7+6Poi8GIhCWmAhP2MIXCY1dYYiY3q9EV3RESLTXIiVaRCMH87aYiR2SnRCXTRwGUcf0fUm/XimqT78ZM4a0RVpc+kQkqYkvFM+AxkNQ5ZP1awXedTpSYLON1/Q0FaZH1ESMiFonl3FuLmfS26tZdUJreILVOrSSEBT2g1Va85CeLv1OaIX8fU2cNo0GJBXRyCKaS1RJYR0RPaXn2TcjRMIAn00jtrrK7iqreT6g95iwDBrHsJRt1cRzFRcfmm3YyjwuYroe8VdaobSvCms4X4ODuMtVMO8xUcbiDY3KyFU6i9m1nitRyLMVupg/AYdP3DFRAmS+Q/5ziKOcqxZhsg4DkbzEWkJkUFshf4/5LJtDimcL9sSImoZ8c6AnVJWV253Z6dLTBHZE8inNjaCHfaGTPTawI2Rv5kUd3BH9muKXBRutBXoblcnx1GlNnF064QY048yxpCekPzOLpHUp0TT1bokfoN+muOvwCue1TrIBLJRZHEOXr2Lfo77CfvHom6ns2YyiQ4cHO0bIljHRcn0SifV9TauIUT+uQk3p6cJrIVB0aGf/TDH2sCaO6AS6pzQeQA9La9Tr2mJf2TEraWXfOOhOy1o15B/XruyMi1pZzkelaiore9DUEY/VijkCbHfRa5bqhrInx1jdbMEqE/bnXYNztGwfS+XxKcdqgGwcQaqMy15cvc5Va9lDneiIJfv+kjh6kBoqeozaDJWmnUL9HagqCRAPC09P5YSsd9Y2XZhJie4pT7nEJ+uc9ySe+YUwDeRUH94rzlchPlPy2W7AHT8FtVruFJVVlo1olM5nquUdjGz4R4444hIvhK9DrOXz+UcrzGyKfTN4/hIiZ2yikDZFHyA3Vq/8WDSpFzQpq8r+5EgzJcLpJKTxwznnQ8Jl3jG0+EThCk/n87PaGBwSZaY8NsRcVoe5DQ463wCcfZV5kkNWVLLAwVVdrZXzclXeSHq17GCe72VZSa+WZcoVctNR/SAG1nmNXII4sIRXyNLjGqkAe4KkS5t/2sA+H3vtROVFpCw8q5H00GPkqgJVlS9q7UvQBSPlE5b5+S08eIX+N8MJaFs/5rLJVt7M5a7fyqe5/HRLz+aS3pb+zSXf1Eja2BucOY1lXtbWFHMd1XLJfKrbE2zaTbLzi6w2Hd8meZH37IHqTJrabzgr5O7YQw/QN+rPy/sOV3K3to4n6L4m+lAEzYOK2jm5MbQ8ghO1l2e7eXYKjwt93oHG/ISe4jQ1VTtCdq5qwZPSF99RT3lBZxUeH4hv6JzUpe/npd1jU1S+r0xU3yni7hHyPvX9Q+wP2+OG6lbRKt09yjqekYbsM1O3tJzbVWfpdUjb6F2NLKM3WDrTL2qSp7YRccmbzHUNYp4Pq/Gq47saWf7jEOAknd9nbgK5eFfbDC/PmfU367r7/vpbaEjPIX276Iyx4j1E1rnokDbVqrwpy1PIwdKdn/N3j6qCczd9cH9BXxl/8RZ5iVvghDhyHnkbSCAtaUXEdffREDKJOnXFuLvlN9R26cYp8RZtiwt55Sm8nupQtuq6Oxe7ze7iv1jLg9O9dvfbdufl4+bTJ+ofrjviM/FAPCI/7YunVJlH5FdL/C7+Fv+Ifxtp47fGH40/JevtW0qmIUqvxl//AzoI3vQ=</latexit><latexit sha1_base64="n+16VmBiHMgfmUjdepzMWj9en4Q=">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</latexit>

Brady Neal⇡) b + ⇡ b⇡aBounds: No-Assumptions Bound(1⇡) a = ba
10 / 48

No-Assumptions Interval Length
Recall trivial bound:<latexit sha1_base64="yhX9WtXCdiGLM3RE0rncm1vlvfw=">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</latexit>ab  E[Y (1)Y (0)]  baTrivial length limit: 2(ba)<latexit sha1_base64="xDUWc/MYNypUpJO9sx67/fWnYkY=">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</latexit>

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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No-assumptions interval length: (1<latexit sha1_base64="n+16VmBiHMgfmUjdepzMWj9en4Q=">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</latexit><latexit sha1_base64="n+16VmBiHMgfmUjdepzMWj9en4Q=">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</latexit>

Brady Neal⇡) b + ⇡ b⇡aBounds: No-Assumptions Bound(1⇡) a = ba
10 / 48

Running Example

Brady Neal

Bounds: No-Assumptions Bound

11 / 48

Running Example
Potential outcomes bounded between 0 (a) and 1 (b)Brady Neal

Bounds: No-Assumptions Bound

11 / 48

Running Example
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

Brady NealE[Y | T = 1] = .9

<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

Bounds: No-Assumptions Bound

11 / 48

Running Example
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

Brady Neal

Bounds: No-Assumptions Bound

11 / 48

Running Example
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">AAAT4XicrVjLbttGFJ2kr8R9OfGyGzZOgC5kRXJTOF0ICGA7KIoGcAq/2sgwJIqyCPNVkrLqCPqAopui235Ft+2X9G967pmhKOpFObAFk6M79557575mRu3Ic5O0Vvvvzt333v/gw4/u3V/7+JNPP/t8/cHD4yTsx7ZzZIdeGJ+2W4njuYFzlLqp55xGsdPy255z0r7clfmTKydO3DA4TK8j58xvXQRu17VbKUjn60+a+2+aYT+1Q9+xmr7bsZop5FPfCVKrYdXP8Kh+e76+WavW+GfNDupmsKnM30H44P6Pqqk6KlS26itfOSpQKcaeaqkEnzeqrmoqAu1MDUGLMXI576iRWoNsH1wOOFqgXuJ5gW9vDDXAd8FMKG1Di4f/GJKWemJ4Ohh3SdVv0W9N8C7SMSS22HiNd9tg+qCmqgdqmVzGubpcGx+/ZNUp1vCcq3WxkogU8YNdWHMXbw/fU6xQntfgdDDqQCrGyAbNA1VTREeMt/a8+KbHSLTI52AkVi+ze5nVMh/icwmsFsYJLRVbLfXSxCWgZoe2Co/HmC5G/BUr1PYtw+qO1+Ay7noVwnsIyqV6i1EReZnOyexazJUa5NEcXcKfkiMAdwJKyMx34T8XHMVctYEpOlqM5AXXEjGDqgb5e85n2Rwhnlu0J2HULOabSz2Rqazc7sxOD+82sWPIDzHXox7xRQX2OMSOmb2ZFyvkjvFtwG82bbSn6FVWpsSzgjVJdlWAG2LGHWNpT2h/ZhZp64agWeazpX6gfgdxr9ArktcVyIa0UGdxQl2BiX0DfUX84uMpVPFsRqlQh087esyWS9ByfRpJ9D3FKhLWj2dQh3h79FpElAq1i38GHPtck0S0T90DjDvUI9IWel1V7Rg7RgWt4huX3WlWq8X8k9rVnXFaq8gFrFTLZGWDmmrqmVmxREDsnvSabbqh7skJVzeasqpN+/OuITlatE+k8vgUY9VhNvYoVcQVL85f57y1bLNOKoyl+P4CHA1KdQ09YW1GRtPaRP3tmioJGQ+bb9/khK530TaYmhmC7htPeeDTdS57ksz8AswWc6pJ703OL0PcN/LZbiAdf0jqcrljVlZRNsZoOJ5ZLu9y5NA/eiQR13gRfR1xLV+O/62JmVWxbwcvmEGUjE0N0qroHebG/JUfqk30gk1kVdGfEmmhxDydRBg/HnM+Bq7wXlJLAIpU+HA8PyqNwR4oI+OxLueyOsxtcNn5OuRsmszTHLqi0ikOqerlWiUv5+WNpi+X7YzzvSir6ctlhXLF3HRNP0iIdVoilzIOIuFPZOlhiVTIPUHTtc0/rWBfwL22b/IiNhaelEj67DF6VaGpylel9qXsgrHxicj8/A4evGL/G/EEdFM/5rLpjbyZy12/k09z+cENPZtL+jf0by75tkTS4d7gjmki87q0poTroJRL51PZnuBgN8nOL7raKny24UXZszumM1lmv5Gs0Ltjgz2gslJ/nt13pJLrpXXcZ/dtsw/F1NxZUjtHt4aWR7Bv9vJsN89O4clEn3epMT+hD3maGpgdITtXbdGT2hffoae8wllFxrvqa5yT6ni+LOweq6LKfaVv+s4k7jaQd9D397g/3Bw3MreKrcLdo6hjHxqy/5G5peXcnjlLL0K6id75yDp6nZkz/bQmfWrrgUvfZK5LEPN8mI+3PL7zkfUvDiFP0vl95jaQJ+9qq+HlObP4Zl123198C43w7uLpsTMmhnePWeexQzqoVX1T1qeQ3Zk7v+TvNqpCcnf46OGUviL+9C3ygrfAPjhyHn0bSCmtaZOIi+6jEWVSc+pKeHfLb6jVwo1T403blkzklW/wGqZDOabrrp2vb9anf8WaHRxvV+vfVGuvn22+eG5+4bqnvlCP1Ffw0456gco8gF9t9bv6W/2j/t2wN37b+GPjT816946R2VCFv42//geLj9wD</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">AAATw3icrVjbbttGEN2kt8S9OdFjX9goBfogq5KTwnkREMB2UBQN4BS24zQOAl0oixBvJSmrjqCv6Gv7Yf2bnjm7FEndKAWWIHE1O3Nmdm67q07oOnHSaPx35+4nn372+Rf37u98+dXX33y7++DheRyMoq591g3cILrotGPbdXz7LHES174II7vtdVz7dWd4KPOvr+0odgL/NLkJ7Xde+8p3+k63nYD05jJ0rJbVqD95v1tt1Bt8WYuDphlUlXmdBA/u/64uVU8FqqtGylO28lWCsavaKsb7rWqqhgpBe6cmoEUYOZy31VTtQHYELhscbVCH+L7Cr7eG6uO3YMaU7kKLi08ESUv9YHh6GPdJ1U/Rb+V4V+mYEFtsvMGzYzA9UBM1ALVMLuXcXK6Dt1ey6gRreMbVOlhJSIr4oVtYcx9PF78TrFC+b8BpY9SDVIRRFzQXVE0RHRGe2vPimwEj0SafjZFYvc7udVbLfID3EFhtjGNaKrZa6oWJi0/NNm0VHpcxXY34F1ao7VuH1Z+twWHc9SqE9xSUofqAURF5nc58dq3mSgzydIku4U/I4YM7BiVg5jvwnwOOYq52gSk62ozkFdcSMoPqBvlXzqfZHCKee7QnZtQs5ptDPaGprMzu1E4Xzw6xI8hPMDegHvFFDfbYxI6YvakXa+SO8GvMX13a2J2j11mZEs8a1iTZVQNugBlnhqU9of2ZWqStm4Bmmfee+o36bcS9Rq9IXtcgG9BCncUxdfkm9i30FfGLh2+himdTSo06PNoxYLYMQcv0aSTR9xNWEbN+XIM6wdOl10Ki1Khd/DPm2OOaJKIj6h5j3KMekbbQ6+rqwNgxLWgV3zjsTotaLeaf1K7ujPNaRc5npVomK1vU1FBPzYolAmJ33mtd0w11T465uumcVR3an3UNydGifSKVxacYqx6zcUCpIq54cfk6l61ln3VSYyzF91fgaFGqb+gxazM0mnZy9XdoqiRgPLp8eiYndL2LtvHczAR0z3jKBZ+uc9mTZOZPYLaZU5f0Xn5+HeKxkU93A+n4E1LXy52zsoqyEUaT2cx6eYcjm/7RI4m4xgvp65Br+X72sXIzm2LfDp6/gCgZmxikTdF7zI3lKz9VVfSCKrKq6E+JtFAink5CjB/POB8DV3iH1OKDIhU+mc1PS2NwBMrUeKzPubQOMxscdr4eOS9N5mkOXVHJHIdU9XqtkpfL8kbT18v2ZvlelNX09bJCuWZuOqYfxMS6KJFLGAeR8HJZeloiFXBP0HRt85sN7PO5145MXkTGwtclkh57jF5VYKryZal9CbtgZHwiMn98hAev2f+mPAFt68dMNtnKm5nczUf5NJMfb+nZTNLb0r+Z5IcSSZt7gzOjicyr0poSrpNSLp1PZXuCjd0kPb/oaqvxuwMvyp7dM53JMvuNZIXeHVvsAbWN+vPiviOV3Cyt4xG7b4d9KKLm3praObs1tCyCI7OXp7t5egqPc33eocbshD7haWpsdoT0XLVHT2pf/IKe8hJnFRkfqic4JzXx/aKwe2yKKveVkek7edx9IB+g7x9xf9geNzS3ir3C3aOo4xga0s/U3NIybtecpVchbaN3ObKOXm/hTD+vSZ/aBuDSN5mbEsQsH5bjrY/vcmT9j0PAk3R2n7kN5PxdbTO8LGdW36zL7vurb6Ehnn18u+yMseE9Yta57JA2alXflPUp5HDhzi/5u4+qkNydPHo4p6+IP3+LvOItcASOjEffBhJKa1oecdV9NKRMYk5dMe9u2Q21Xrhxarx52+JcXnkGr2U6lG267s773Wpz/l+sxcH5fr35c73x6mn1+TPzD9c99Z16pH6Enw7Uc1TmCfwqN66/1T/q38pxZViJKolmvXvHyFRU4VWZ/g8Zn9C0</latexit>

E[Y | T = 1] = .9

<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit><latexit sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

Plugging these in to the no-assumptions bound:E[Y (1)<latexit sha1_base64="HbJtj2tsaFvjiI4eMSWUKiAZIAs=">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</latexit>

E[Y (1)
Brady Neal

Y (0)]  (.3)(.9) + (1Y (0)](.3)(.9) + (1.3)(1)(.3)(0)(1.3)(.2).3)(0)(.3)(1)(1.3)(.2)

Bounds: No-Assumptions Bound

11 / 48

Running Example
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

No-Assumptions BoundE[Y (1)

Y (0)]  ⇡ E[Y | T = 1] + (1

⇡) b

⇡a

(1

⇡) E[Y | T = 0]

E[Y (1)

Y (0)]⇡ E[Y | T = 1] + (1

⇡) a

⇡b

(1

⇡) E[Y | T = 0]<latexit 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sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit><latexit sha1_base64="owBS+mkdMoFuagOnONlY57Ix4D0=">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</latexit>

Plugging these in to the no-assumptions bound:E[Y (1)<latexit sha1_base64="HbJtj2tsaFvjiI4eMSWUKiAZIAs=">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</latexit>

E[Y (1)

Y (0)]  (.3)(.9) + (1Y (0)]<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit>

Brady Neal

.3)(1)(.3)(0)(1.3)(.2)(.3)(.9) + (1 .3)(0) (.3)(1)
0.17  E[Y (1) Y (0)]  0.83

(1.3)(.2)

Bounds: No-Assumptions Bound

11 / 48

Questions:
1. What kind of bounds can we get on the ATE if thepotential outcomes are unbounded?
2. Assuming bounded potential outcomes, how muchsmaller of an interval can we get than the trivial interval[a – b, b – a]?
3. Re-derive the Observational-Counterfactual
Decomposition.
4. Derive a more general no-assumptions bound wherea1  Y (1)  b1 and a0  Y (0)  b0 .
<latexit sha1_base64="OKXTAo1N/HuabFZWDTvYkrPCPOA=">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</latexit>

Bounds
No-Assumptions Bound
Monotone Treatment Response
Monotone Treatment Selection
Optimal Treatment Selection

Sensitivity Analysis
Linear Single Confounder
Towards More General Settings

Brady Neal

Bounds: Monotone Treatment Response

13 / 48

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">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</latexit>

Brady Neal

Bounds: Monotone Treatment Response

Yi (0)

14 / 48

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">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</latexit>

Yi (0)

Means every ITE is nonnegative, so ITE lower bound comes up from a – b to 0

Brady Neal

Bounds: Monotone Treatment Response

14 / 48

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">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</latexit>

Yi (0)

Means every ITE is nonnegative, so ITE lower bound comes up from a – b to 0
Intuitively, the ATE lower bound should raise up to 0 too<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

Brady NealE[Y (1)

Y (0)]

0

Bounds: Monotone Treatment Response

14 / 48

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">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</latexit>

Yi (0)

Means every ITE is nonnegative, so ITE lower bound comes up from a – b to 0
Intuitively, the ATE lower bound should raise up to 0 too<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

E[Y (1)

Y (0)]

0

Proof:<latexit sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]

⇡ E[Y | T = 1] + (1=0
Brady Neal

⇡ E[Y | T = 1]

(1⇡) E[Y (1) | T = 0](1⇡) E[Y | T = 0]

(Observational-Counterfactual
Decomposition)⇡) E[Y | T = 0]⇡) E[Y | T = 0]

Bounds: Monotone Treatment Response

14 / 48

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">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</latexit>

Yi (0)

Means every ITE is nonnegative, so ITE lower bound comes up from a – b to 0
Intuitively, the ATE lower bound should raise up to 0 too<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

E[Y (1)

Y (0)]

0

Proof:<latexit sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">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</latexit><latexit sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">AAAVkHicrVhZb9tGEN64V+Jetgs/9WUbpUGCyKrkpnBQQGgK20ERNIBTOHHa0HAlcmWz4mWSsusI+pV96r/pzLdLkdRFObUEi8vZmW9m59pddyPPTdJm899bKx98+NHHn9y+s/rpZ59/8eXa+sbrJBzEtnplh14Yv+l2EuW5gXqVuqmn3kSx6vhdTx11+7s8f3Sh4sQNg8P0KlLHfuc0cHuu3UmJdLJ+6x9r/60VDlI79NWD1kO5JcdvzYfH8n5bWpErrbos8EnLdx1ppaQo9VWQyrZsHctH8kGLxSP34QQ/406LNI+lFYTBwO+qWFrWqqTPfet80HE0yiRIcxYI6d2aq3eO0kzVqTq/gcX9/5Xd7LLo5WSt1mw08ZHTg5YZ1IT5HITrd34TlnBEKGwxEL5QIhApjT3REQl934qWaIqIaMdiSLSYRi7mlRiJVZIdEJcijg5R+/R7Sm9vDTWgd8ZMIG2TFo/+YpKU4lvD49C4B6p+sn5Z4J2nYwhstvGKnl2D6RM1FWdErZLLOJeX69LXr1h1Smt4gtW6tJIIFPaDXVpzj54evae0Qv69Ik5FI4ekYhrZRPOIqimsI6an9jz75gyR6IBP0YitXmT3Iqt5PqRvn7A6NE5gKdsqxTMTlwCaFWxlHg8xnY/4N61Q27cIqzdeg4u461Uw7yFR+uIdjcrIi3QWs2s+V2qQRzN0MX8KjoC4E6KEyHyX/OcSRzlXbcJkHR1E8hRriZBBDYP8HPNZNkcUzy3YkyBqEvnmQk9kKiu3O7PTo2cX2DHJD2nuDHrYF3WyRwE7RvZmXqyDO6a3S7zZsNGeoDdQmRzPOq2Js6tOuCHNuGMs7Qntz8wibd2QaNJ8t8Sv0K8o7nV4hfO6TrIhLNRZnEBXYGLfpr7CfvHpl6ns2YxShw4fdpwhW/pEy/VpJNb3Ha0iQf14BnVITw9ei4BSh3b2zyXGPtbEER1A9yWNHehhaUm9riF2jB2jklb2jYvuNK1VIv+4dnVnnNTKcgEqVZqsbENTUzw2K+YIsN1Fr9mmG+qenGB1owmrurA/7xqco2X7WCqPTzlWDrLxDFJlXPbi7HXOWss26qSOWLLvT4mjDameoSeozchoWi3U366pkhDxsPH0TU7oemdtlxMzQ6L7xlMe8ek65z2JZ84Js4OcsuC94vwixH0jn+0G3PGHoC6We43KKsvGNBqOZxbLuxgp+EePOOIaL4KvI6zlm/GfLMwsi30zeMEUImdsapCWRXeQG7NXfihq1AtqlFVlf3KkmRLjdBLR+N6Y8x7hMm8fWgKicIUPx/OjyhjsEWVkPNbDXFaHuQ0uOp8DTstknubQFZVOcHBVL9bKeTkrbzR9sawzzveyrKYvlmXKBXLTNf0gAdabCrkUcWAJv5ClhxVSIfYETdc2/76EfQH22oHJi9hYeFQh6aPH6FWFpipfVNqXogvGxics88d7ePAC/W+EE9B1/ZjLptfyZi539V4+zeUvr+nZXNK/pn9zyXcVkgp7gzumsczLyppiroNKLp1PVXuCot0kO7/oaqvjt0te5D3bMZ1Jmv2Gs0Lvjm30gPpS/Xl63+FKblXW8QDdt4s+FEOzs6B2Xt0YWh7BgdnLs908O4UnhT7vQmN+Qh/iNHVpdoTsXLUFT2pf/EI95QWdVXi8K76nc1KLfp+Vdo9lUfm+MjB9p4i7Tcg71Pf3sD9cHzcyt4qt0t2jrGOfNGR/I3NLy7k9c5aeh3QdvbORdfScqTP9pCZ9ajsjLn2TuapAzPNhNt7i+M5G1v9xCHGSzu8zN4FcvKsth5fnzPybddV9f/4tNKJnj349dMbE8O4h6zx0SEW1qm/K+hSyO3Xn5/zdpqrg3B3e3ZjQV8afvEWe4hY4II6cR98GUkhrWhFx3n00gkxqTl0J7m75DbVRunFqvEnbkkJe+QavbTqUMl139WSt1pr8L9b04PV2o/VDo/nyce3pE/Mfrtvia3FXPCA/7YinVJkH5Fd75ceVP1fclb82NzafbP60+bNmXbllZL4Spc/m8/8AIrJeWw==</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

⇡ E[Y | T = 1] + (1 ⇡) E[Y | T = 0]
⇡ E[Y | T = 1] (1 ⇡) E[Y | T = 0]=0
Brady Neal

Bounds: Monotone Treatment Response

14 / 48

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">AAAT6XicrVjbbttGEN2kt8S9OdFjX9goBRJAViU3hQu0BgLYCYqiAZzCtzYKDImiZEKkyJCUXUfQRxR9Kfrar+hr+xX9m545uxRJ3SgHkSBxNTtzZnZuu6tO6Llx0mj8d+PmO+++9/4Ht25vfPjRx598unnn7nEcjCLbObIDL4hOO+3Y8dyhc5S4ieechpHT9juec9IZ7Mn8yYUTxW4wPEyuQuel3+4P3Z5rtxOQzja3Wr0ganue5Vqtb61WMErswHfO3AfNh1ar77zKkxoPzzarjXqDL2t+0DSDqjKvg+DO7Z9US3VVoGw1Ur5y1FAlGHuqrWK8X6imaqgQtJdqDFqEkct5R03UBmRH4HLA0QZ1gO8+fr0w1CF+C2ZMaRtaPHwiSFrqC8PTxbhHqn6KfivHu0zHmNhi4xWeHYPpg5qoc1DL5FLO9eU6ePslq06whm+4WhcrCUkRP9iFNffw9PA7wQrl+wqcDkZdSEUY2aB5oGqK6Ijw1J4X35wzEm3yORiJ1avsXmW1zAd4D4DVxjimpWKrpZ6auAyp2aGtwuMxpssRf8UKtX2rsHrTNbiMu16F8B6CMlCvMSoir9KZz67lXIlBnizQJfwJOYbgjkEJmPku/OeCo5irNjBFR5uR7HMtITOobpB/4HyazSHiuUV7YkbNYr651BOaysrsTu308OwQO4L8GHPn1CO+qMEeh9gRszf1Yo3cEX5d8pdNG+0Zep2VKfGsYU2SXTXgBphxp1jaE9qfqUXaujFolnlvqR+p30Hca/SK5HUNsgEt1FkcU9fQxH4XfUX84uNbqOLZlFKjDp92nDNbBqBl+jSS6PsSq4hZP55BHePp0WshUWrULv655NjnmiSiI+q+xLhLPSJtodfV1Y6xY1LQKr5x2Z3mtVrMP6ld3RlntYrckJVqmazcpaaGemRWLBEQu/Nes0031D055uomM1Z1aH/WNSRHi/aJVBafYqy6zMZzShVxxYuL17loLduskxpjKb7vg2OXUj1Dj1mbodG0kau/PVMlAeNh8+mbnND1LtouZ2bGoPvGUx74dJ3LniQzr4DZZk616L38/CrEJ0Y+3Q2k449JXS13zMoqykYYjaczq+Vdjhz6R48k4hovpK9DruXz6cfKzayL/XbwhnOIkrGJQVoXvcvcWLzyQ1VFL6giq4r+lEgLJeLpJMT4/pTzPnCFd0AtQ1CkwsfT+UlpDPZBmRiP9TiX1mFmg8vO1yVny2Se5tAVlcxwSFWv1ip5uShvNH21bHea70VZTV8tK5QL5qZr+kFMrNMSuYRxEAk/l6WHJVIB9wRN1zb/vIZ9Q+61I5MXkbHwpETSZ4/RqwpMVT4rtS9hF4yMT0Tmlzfw4AX734QnoOv6MZNNruXNTO7qjXyayV9e07OZpH9N/2aSr0skHe4N7pQmMs9La0q4Dkq5dD6V7QkOdpP0/KKrrcbvDrwoe3bXdCbL7DeSFXp33GUPqK3Vn+f3HankZmkdj9h9O+xDETV3V9TO0VtDyyI4Mnt5upunp/A41+ddasxO6GOepi7NjpCeq7boSe2L79FTnuGsIuM99RXOSU18Py3sHuuiyn1lZPpOHncbyDvo+/vcH66PG5pbxVbh7lHU8QQa0s/E3NIybs+cpZchXUfvYmQdve7cmX5Wkz61nYNL32SuShCzfFiMtzq+i5H1Pw4BT9LZfeZtIOfvauvhZTmz/GZddt9ffgsN8ezh22NnjA3vPrPOY4d0UKv6pqxPIXtzd37J321UheTu+N7dGX1F/NlbZJ+3wBE4Mh59G0gorWl5xGX30ZAyiTl1xby7ZTfUeuHGqfFmbYtzeeUbvF3ToRzTdTfONqvN2X+x5gfH2/Xm1/XG80fVx9+Zf7huqc/UPfUAftpRj1GZB/CrrX5Xf6t/1L+VQeW3yh+VPzXrzRtGpqIKr8pf/wPVt99m</latexit>

Yi (0)

Means every ITE is nonnegative, so ITE lower bound comes up from a – b to 0
Intuitively, the ATE lower bound should raise up to 0 too<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

E[Y (1)

Y (0)]

0

Proof:<latexit sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">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</latexit><latexit sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

⇡ E[Y | T = 1] + (1 ⇡) E[Y | T = 0]
⇡ E[Y | T = 1] (1 ⇡) E[Y | T = 0]=0
Brady Neal

Bounds: Monotone Treatment Response

14 / 48

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">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</latexit>

Yi (0)

Means every ITE is nonnegative, so ITE lower bound comes up from a – b to 0
Intuitively, the ATE lower bound should raise up to 0 too<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

E[Y (1)

Y (0)]

0

Proof:<latexit sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">AAAVkHicrVhZb9tGEN64V+Jetgs/9WUbpUGCyKrkpnBQQGgK20ERNIBTOHHa0HAlcmWz4mWSsusI+pV96r/pzLdLkdRFObUEi8vZmW9m59pddyPPTdJm899bKx98+NHHn9y+s/rpZ59/8eXa+sbrJBzEtnplh14Yv+l2EuW5gXqVuqmn3kSx6vhdTx11+7s8f3Sh4sQNg8P0KlLHfuc0cHuu3UmJdLJ+6x9r/60VDlI79NWD1kO5JcdvzYfH8n5bWpErrbos8EnLdx1ppaQo9VWQyrZsHctH8kGLxSP34QQ/406LNI+lFYTBwO+qWFrWqqTPfet80HE0yiRIcxYI6d2aq3eO0kzVqTq/gcX9/5Xd7LLo5WSt1mw08ZHTg5YZ1IT5HITrd34TlnBEKGwxEL5QIhApjT3REQl934qWaIqIaMdiSLSYRi7mlRiJVZIdEJcijg5R+/R7Sm9vDTWgd8ZMIG2TFo/+YpKU4lvD49C4B6p+sn5Z4J2nYwhstvGKnl2D6RM1FWdErZLLOJeX69LXr1h1Smt4gtW6tJIIFPaDXVpzj54evae0Qv69Ik5FI4ekYhrZRPOIqimsI6an9jz75gyR6IBP0YitXmT3Iqt5PqRvn7A6NE5gKdsqxTMTlwCaFWxlHg8xnY/4N61Q27cIqzdeg4u461Uw7yFR+uIdjcrIi3QWs2s+V2qQRzN0MX8KjoC4E6KEyHyX/OcSRzlXbcJkHR1E8hRriZBBDYP8HPNZNkcUzy3YkyBqEvnmQk9kKiu3O7PTo2cX2DHJD2nuDHrYF3WyRwE7RvZmXqyDO6a3S7zZsNGeoDdQmRzPOq2Js6tOuCHNuGMs7Qntz8wibd2QaNJ8t8Sv0K8o7nV4hfO6TrIhLNRZnEBXYGLfpr7CfvHpl6ns2YxShw4fdpwhW/pEy/VpJNb3Ha0iQf14BnVITw9ei4BSh3b2zyXGPtbEER1A9yWNHehhaUm9riF2jB2jklb2jYvuNK1VIv+4dnVnnNTKcgEqVZqsbENTUzw2K+YIsN1Fr9mmG+qenGB1owmrurA/7xqco2X7WCqPTzlWDrLxDFJlXPbi7HXOWss26qSOWLLvT4mjDameoSeozchoWi3U366pkhDxsPH0TU7oemdtlxMzQ6L7xlMe8ek65z2JZ84Js4OcsuC94vwixH0jn+0G3PGHoC6We43KKsvGNBqOZxbLuxgp+EePOOIaL4KvI6zlm/GfLMwsi30zeMEUImdsapCWRXeQG7NXfihq1AtqlFVlf3KkmRLjdBLR+N6Y8x7hMm8fWgKicIUPx/OjyhjsEWVkPNbDXFaHuQ0uOp8DTstknubQFZVOcHBVL9bKeTkrbzR9sawzzveyrKYvlmXKBXLTNf0gAdabCrkUcWAJv5ClhxVSIfYETdc2/76EfQH22oHJi9hYeFQh6aPH6FWFpipfVNqXogvGxics88d7ePAC/W+EE9B1/ZjLptfyZi539V4+zeUvr+nZXNK/pn9zyXcVkgp7gzumsczLyppiroNKLp1PVXuCot0kO7/oaqvjt0te5D3bMZ1Jmv2Gs0Lvjm30gPpS/Xl63+FKblXW8QDdt4s+FEOzs6B2Xt0YWh7BgdnLs908O4UnhT7vQmN+Qh/iNHVpdoTsXLUFT2pf/EI95QWdVXi8K76nc1KLfp+Vdo9lUfm+MjB9p4i7Tcg71Pf3sD9cHzcyt4qt0t2jrGOfNGR/I3NLy7k9c5aeh3QdvbORdfScqTP9pCZ9ajsjLn2TuapAzPNhNt7i+M5G1v9xCHGSzu8zN4FcvKsth5fnzPybddV9f/4tNKJnj349dMbE8O4h6zx0SEW1qm/K+hSyO3Xn5/zdpqrg3B3e3ZjQV8afvEWe4hY4II6cR98GUkhrWhFx3n00gkxqTl0J7m75DbVRunFqvEnbkkJe+QavbTqUMl139WSt1pr8L9b04PV2o/VDo/nyce3pE/Mfrtvia3FXPCA/7YinVJkH5Fd75ceVP1fclb82NzafbP60+bNmXbllZL4Spc/m8/8AIrJeWw==</latexit><latexit sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

⇡ E[Y | T = 1] + (1 ⇡) E[Y | T = 0]
⇡ E[Y | T = 1] (1 ⇡) E[Y | T = 0]<latexit sha1_base64="Hov6kO8KBesbhq2/+bffS7dv/tk=">AAAUK3icrVhLb9tGEF6nSZu4Lyc69sJGKZACsiq5KdyLgAC2g6JoAKfwq7UMQyJXEiG+QlJWHUE/qb+m6KFFr/0VvXTm26VI6kFKgSVIXM3OfDM7r91VN3DsKG40/tq698H9Bx9+9PDR9seffPrZ5zuPn5xF/ig05anpO3540e1E0rE9eRrbsSMvglB23K4jz7vDA54/v5FhZPveSXwbyCu30/fsnm12YiJd7/jto8u2P4pN35XPm18bbde2jHZMELErvdhoGY0ro92Xb40sY2MFYyvLtZTleqfaqDfwMhYHTT2oCv069h8/+lm0hSV8YYqRcIUUnohp7IiOiOh9KZqiIQKiXYkJ0UIa2ZiXYiq2SXZEXJI4OkQd0neffl1qqke/GTOCtElaHPqEJGmIrzSPReMeqOrJ+o0M7yodE2Czjbf07GpMl6ixGBC1TC7hXF+uS2+3ZNUxreF7rNamlQSgsB/M3Jp79HTod0wr5O9b4pQ0skgqpJFJNIeoisI6Qnoqz7NvBohEB3ySRmx1kd1FVvO8T+8hYXVoHMFSttUQr3RcPGiWsJV5HMR0NeJvtEJlXxFWb7YGG3FXq2DeE6IMxTsa5ZGLdGazazVXrJGnS3QxfwwOj7gjovjIfJv8ZxNHPldNwmQdHUSyj7UEyKC6Rv4R80k2BxTPXdgTIWoG8s2GnkBXVmp3YqdDzy6wQ5Kf0NwAetgXNbJHAjtE9iZerIE7pF9j/DJhozlHr6MyOZ41WhNnV41wfZqxZ1jKE8qfiUXKugnRDP3eFT9Bv6S41+AVzusayfqwUGVxBF2ejn2L+gr7xaVvprJnE0oNOlzYMUC2DImW6lNIrO8bWkWE+nE06oSeDrwWAKUG7eyfMcYu1sQRHUH3mMYW9LC0Qb2uLva1HdOcVvaNje60qNVA/nHtqs44r5XlPFSqobOyBU0N8UKvmCPAdme9ZupuqHpyhNVN56zqwv60a3CO5u1jqTQ++VhZyMYBpPK47MXl61y2lj3USQ2xZN/3iaMFqZ6mR6jNQGvaztTfga4SH/Ew8XR1Tqh6Z23juZkJ0V3tKYf4VJ3znsQzbwmzg5xqw3vZ+SLEIy2f7Abc8SegFsudobLysiGNJrOZYnkbIwn/qBFHXOEF8HWAtXw5+xiZmXWx7wbPW0DkjI010rroFnJj+cpPRJV6QZWyKu9PjjRTQpxOAho/m3E+I1zmHUKLRxSu8Mlsfloag0OiTLXHephL6jC1wUbns8DZ1pmnOFRFxXMcXNXFWjkvl+WNohfLWrN8z8sqerEsU26Qm7buBxGwLkrkYsSBJdxMlp6USPnYExRd2fzLGvZ52GtHOi9CbeF5iaSLHqNW5euqfF1qX4wuGGqfsMyv7+HBG/S/KU5Am/oxlY038mYqd/tePk3lxxt6NpV0N/RvKvmuRFJib7BnNJZ5U1pTzHVcyqXyqWxPkLSbJOcXVW01fHfJi7xnW7ozGXq/4axQu2MLPaC2Vn9e3He4kpuldTxC9+2iD4XQbBXUzumdoaURHOm9PNnNk1N4lOnzNjSmJ/QJTlNjvSMk56pdeFL54gfqKa/prMLjA/EtnZOa9P0qt3usi8r3lZHuO1ncPULep75/iP1hc9xA3yp2c3ePvI4j0pB8pvqWlnI7+iy9CmkTvcuRVfSshTP9vCZ1ahsQl7rJ3JYgpvmwHK84vsuR1T8OPk7S6X3mLpCzd7X18NKcWX2zLrvvr76FBvTs0beDzhhp3kNknYMOKalW1U1ZnUIOFu78nL97VBWcu5OnT+b05fHnb5F93AJHxJHyqNtADGlFyyKuuo8GkIn1qSvC3S29odZzN06FN29blMkrV+O1dIeSuutuX+9Um/P/Yi0Ozvbqze/qjTcvqi8b+h+uh+IL8VQ8Jz/ti5dUmcfkV1P8Kf7bur/1oPJ75Y/K35V/FOu9LS1TEblX5d//AYQH9h0=</latexit><latexit sha1_base64="Hov6kO8KBesbhq2/+bffS7dv/tk=">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</latexit>

E[Y (1) | T = 0]

E[Y (0) | T = 0] = E=0Brady Neal

Bounds: Monotone Treatment Response

14 / 48

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">AAAT6XicrVjbbttGEN2kt8S9OdFjX9goBRJAViU3hQu0BgLYCYqiAZzCtzYKDImiZEKkyJCUXUfQRxR9Kfrar+hr+xX9m545uxRJ3SgHkSBxNTtzZnZuu6tO6Llx0mj8d+PmO+++9/4Ht25vfPjRx598unnn7nEcjCLbObIDL4hOO+3Y8dyhc5S4ieechpHT9juec9IZ7Mn8yYUTxW4wPEyuQuel3+4P3Z5rtxOQzja3Wr0ganue5Vqtb61WMErswHfO3AfNh1ar77zKkxoPzzarjXqDL2t+0DSDqjKvg+DO7Z9US3VVoGw1Ur5y1FAlGHuqrWK8X6imaqgQtJdqDFqEkct5R03UBmRH4HLA0QZ1gO8+fr0w1CF+C2ZMaRtaPHwiSFrqC8PTxbhHqn6KfivHu0zHmNhi4xWeHYPpg5qoc1DL5FLO9eU6ePslq06whm+4WhcrCUkRP9iFNffw9PA7wQrl+wqcDkZdSEUY2aB5oGqK6Ijw1J4X35wzEm3yORiJ1avsXmW1zAd4D4DVxjimpWKrpZ6auAyp2aGtwuMxpssRf8UKtX2rsHrTNbiMu16F8B6CMlCvMSoir9KZz67lXIlBnizQJfwJOYbgjkEJmPku/OeCo5irNjBFR5uR7HMtITOobpB/4HyazSHiuUV7YkbNYr651BOaysrsTu308OwQO4L8GHPn1CO+qMEeh9gRszf1Yo3cEX5d8pdNG+0Zep2VKfGsYU2SXTXgBphxp1jaE9qfqUXaujFolnlvqR+p30Hca/SK5HUNsgEt1FkcU9fQxH4XfUX84uNbqOLZlFKjDp92nDNbBqBl+jSS6PsSq4hZP55BHePp0WshUWrULv655NjnmiSiI+q+xLhLPSJtodfV1Y6xY1LQKr5x2Z3mtVrMP6ld3RlntYrckJVqmazcpaaGemRWLBEQu/Nes0031D055uomM1Z1aH/WNSRHi/aJVBafYqy6zMZzShVxxYuL17loLduskxpjKb7vg2OXUj1Dj1mbodG0kau/PVMlAeNh8+mbnND1LtouZ2bGoPvGUx74dJ3LniQzr4DZZk616L38/CrEJ0Y+3Q2k449JXS13zMoqykYYjaczq+Vdjhz6R48k4hovpK9DruXz6cfKzayL/XbwhnOIkrGJQVoXvcvcWLzyQ1VFL6giq4r+lEgLJeLpJMT4/pTzPnCFd0AtQ1CkwsfT+UlpDPZBmRiP9TiX1mFmg8vO1yVny2Se5tAVlcxwSFWv1ip5uShvNH21bHea70VZTV8tK5QL5qZr+kFMrNMSuYRxEAk/l6WHJVIB9wRN1zb/vIZ9Q+61I5MXkbHwpETSZ4/RqwpMVT4rtS9hF4yMT0Tmlzfw4AX734QnoOv6MZNNruXNTO7qjXyayV9e07OZpH9N/2aSr0skHe4N7pQmMs9La0q4Dkq5dD6V7QkOdpP0/KKrrcbvDrwoe3bXdCbL7DeSFXp33GUPqK3Vn+f3HankZmkdj9h9O+xDETV3V9TO0VtDyyI4Mnt5upunp/A41+ddasxO6GOepi7NjpCeq7boSe2L79FTnuGsIuM99RXOSU18Py3sHuuiyn1lZPpOHncbyDvo+/vcH66PG5pbxVbh7lHU8QQa0s/E3NIybs+cpZchXUfvYmQdve7cmX5Wkz61nYNL32SuShCzfFiMtzq+i5H1Pw4BT9LZfeZtIOfvauvhZTmz/GZddt9ffgsN8ezh22NnjA3vPrPOY4d0UKv6pqxPIXtzd37J321UheTu+N7dGX1F/NlbZJ+3wBE4Mh59G0gorWl5xGX30ZAyiTl1xby7ZTfUeuHGqfFmbYtzeeUbvF3ToRzTdTfONqvN2X+x5gfH2/Xm1/XG80fVx9+Zf7huqc/UPfUAftpRj1GZB/CrrX5Xf6t/1L+VQeW3yh+VPzXrzRtGpqIKr8pf/wPVt99m</latexit>

Yi (0)

Means every ITE is nonnegative, so ITE lower bound comes up from a – b to 0
Intuitively, the ATE lower bound should raise up to 0 too<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

E[Y (1)

Y (0)]

0

Proof:<latexit sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">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</latexit><latexit sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

⇡ E[Y | T = 1] + (1 ⇡) E[Y⇡ E[Y | T = 1] (1 ⇡) E[Y | T = 0]

(1) | T = 0]
| T E[Y= 0]<latexit sha1_base64="Hov6kO8KBesbhq2/+bffS7dv/tk=">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</latexit><latexit sha1_base64="Hov6kO8KBesbhq2/+bffS7dv/tk=">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</latexit>(0) | T = 0] = E[Y (0)| T |=
E[Y (1)
T 0]= 0] = E=0
Brady Neal

Bounds: Monotone Treatment Response

14 / 48

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">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</latexit>

Yi (0)

Means every ITE is nonnegative, so ITE lower bound comes up from a – b to 0
Intuitively, the ATE lower bound should raise up to 0 too<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

E[Y (1)

Y (0)]

0

Proof:<latexit sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">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</latexit><latexit 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E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

⇡ E[Y | T = 1] + (1 ⇡) E[Y⇡ E[Y | T = 1] (1 ⇡) E[Y | T = 0]

(1) | T = 0]
| T E[Y= 0]<latexit sha1_base64="Hov6kO8KBesbhq2/+bffS7dv/tk=">AAAUK3icrVhLb9tGEF6nSZu4Lyc69sJGKZACsiq5KdyLgAC2g6JoAKfwq7UMQyJXEiG+QlJWHUE/qb+m6KFFr/0VvXTm26VI6kFKgSVIXM3OfDM7r91VN3DsKG40/tq698H9Bx9+9PDR9seffPrZ5zuPn5xF/ig05anpO3540e1E0rE9eRrbsSMvglB23K4jz7vDA54/v5FhZPveSXwbyCu30/fsnm12YiJd7/jto8u2P4pN35XPm18bbde2jHZMELErvdhoGY0ro92Xb40sY2MFYyvLtZTleqfaqDfwMhYHTT2oCv069h8/+lm0hSV8YYqRcIUUnohp7IiOiOh9KZqiIQKiXYkJ0UIa2ZiXYiq2SXZEXJI4OkQd0neffl1qqke/GTOCtElaHPqEJGmIrzSPReMeqOrJ+o0M7yodE2Czjbf07GpMl6ixGBC1TC7hXF+uS2+3ZNUxreF7rNamlQSgsB/M3Jp79HTod0wr5O9b4pQ0skgqpJFJNIeoisI6Qnoqz7NvBohEB3ySRmx1kd1FVvO8T+8hYXVoHMFSttUQr3RcPGiWsJV5HMR0NeJvtEJlXxFWb7YGG3FXq2DeE6IMxTsa5ZGLdGazazVXrJGnS3QxfwwOj7gjovjIfJv8ZxNHPldNwmQdHUSyj7UEyKC6Rv4R80k2BxTPXdgTIWoG8s2GnkBXVmp3YqdDzy6wQ5Kf0NwAetgXNbJHAjtE9iZerIE7pF9j/DJhozlHr6MyOZ41WhNnV41wfZqxZ1jKE8qfiUXKugnRDP3eFT9Bv6S41+AVzusayfqwUGVxBF2ejn2L+gr7xaVvprJnE0oNOlzYMUC2DImW6lNIrO8bWkWE+nE06oSeDrwWAKUG7eyfMcYu1sQRHUH3mMYW9LC0Qb2uLva1HdOcVvaNje60qNVA/nHtqs44r5XlPFSqobOyBU0N8UKvmCPAdme9ZupuqHpyhNVN56zqwv60a3CO5u1jqTQ++VhZyMYBpPK47MXl61y2lj3USQ2xZN/3iaMFqZ6mR6jNQGvaztTfga4SH/Ew8XR1Tqh6Z23juZkJ0V3tKYf4VJ3znsQzbwmzg5xqw3vZ+SLEIy2f7Abc8SegFsudobLysiGNJrOZYnkbIwn/qBFHXOEF8HWAtXw5+xiZmXWx7wbPW0DkjI010rroFnJj+cpPRJV6QZWyKu9PjjRTQpxOAho/m3E+I1zmHUKLRxSu8Mlsfloag0OiTLXHephL6jC1wUbns8DZ1pmnOFRFxXMcXNXFWjkvl+WNohfLWrN8z8sqerEsU26Qm7buBxGwLkrkYsSBJdxMlp6USPnYExRd2fzLGvZ52GtHOi9CbeF5iaSLHqNW5euqfF1qX4wuGGqfsMyv7+HBG/S/KU5Am/oxlY038mYqd/tePk3lxxt6NpV0N/RvKvmuRFJib7BnNJZ5U1pTzHVcyqXyqWxPkLSbJOcXVW01fHfJi7xnW7ozGXq/4axQu2MLPaC2Vn9e3He4kpuldTxC9+2iD4XQbBXUzumdoaURHOm9PNnNk1N4lOnzNjSmJ/QJTlNjvSMk56pdeFL54gfqKa/prMLjA/EtnZOa9P0qt3usi8r3lZHuO1ncPULep75/iP1hc9xA3yp2c3ePvI4j0pB8pvqWlnI7+iy9CmkTvcuRVfSshTP9vCZ1ahsQl7rJ3JYgpvmwHK84vsuR1T8OPk7S6X3mLpCzd7X18NKcWX2zLrvvr76FBvTs0beDzhhp3kNknYMOKalW1U1ZnUIOFu78nL97VBWcu5OnT+b05fHnb5F93AJHxJHyqNtADGlFyyKuuo8GkIn1qSvC3S29odZzN06FN29blMkrV+O1dIeSuutuX+9Um/P/Yi0Ozvbqze/qjTcvqi8b+h+uh+IL8VQ8Jz/ti5dUmcfkV1P8Kf7bur/1oPJ75Y/K35V/FOu9LS1TEblX5d//AYQH9h0=</latexit><latexit sha1_base64="Hov6kO8KBesbhq2/+bffS7dv/tk=">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</latexit>(0) | T = 0] = E[Y (0)| T |=
E[Y (1)
T 0]= 0] = E=0
Brady Neal

Bounds: Monotone Treatment Response

14 / 48

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">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</latexit>

Yi (0)

Means every ITE is nonnegative, so ITE lower bound comes up from a – b to 0
Intuitively, the ATE lower bound should raise up to 0 too<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

E[Y (1)

Y (0)]

0

Proof:<latexit sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">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</latexit><latexit 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E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

⇡ E[Y | T = 1] + (1 ⇡) E[Y⇡ E[Y | T = 1] (1 ⇡) E[Y | T = 0]

(1) | T = 0]
| T E[Y= 0]<latexit sha1_base64="Hov6kO8KBesbhq2/+bffS7dv/tk=">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</latexit><latexit sha1_base64="Hov6kO8KBesbhq2/+bffS7dv/tk=">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</latexit>(0) | T = 0] = E[Y (0)| T |=
E[Y (1)
T 0]= 0] = E=0
Brady Neal

Bounds: Monotone Treatment Response

14 / 48

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">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</latexit>

Yi (0)

Means every ITE is nonnegative, so ITE lower bound comes up from a – b to 0
Intuitively, the ATE lower bound should raise up to 0 too<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

E[Y (1)

Y (0)]

0

Proof:<latexit sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">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</latexit><latexit 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E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

⇡ E[Y | T = 1] + (1 ⇡) E[Y⇡ E[Y | T = 1] (1 ⇡) E[Y | T = 0]

(1) | T = 0]
| T E[Y= 0]<latexit 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sha1_base64="aVbYY5WnaeXeuQLuUXKzGfBEDJc=">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</latexit>(0) | T = 0] = E[Y (0)| T |=
E[Y (1)
T 0]= 0] = E

E[Y (1) | T = 1]

E[Y (0) | T = 1]

=0
Brady Neal

Bounds: Monotone Treatment Response

14 / 48

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">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</latexit>

Yi (0)

Means every ITE is nonnegative, so ITE lower bound comes up from a – b to 0
Intuitively, the ATE lower bound should raise up to 0 too<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

E[Y (1)

Y (0)]

0

Proof:<latexit 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E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

⇡ E[Y | T = 1] + (1 ⇡) E[Y⇡ E[Y | T = 1] (1 ⇡) E[Y | T = 0]

(1) | T = 0]
| T E[Y= 0]<latexit 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| T = 0] = E[Y (0)| T |=
E[Y (1)
T 0]= 0] = E

E[Y (0) | T = 1]

E[Y (1) | T = 1] ==0
Brady Neal

Bounds: Monotone Treatment Response

14 / 48

E

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">AAAT6XicrVjbbttGEN2kt8S9OdFjX9goBRJAViU3hQu0BgLYCYqiAZzCtzYKDImiZEKkyJCUXUfQRxR9Kfrar+hr+xX9m545uxRJ3SgHkSBxNTtzZnZuu6tO6Llx0mj8d+PmO+++9/4Ht25vfPjRx598unnn7nEcjCLbObIDL4hOO+3Y8dyhc5S4ieechpHT9juec9IZ7Mn8yYUTxW4wPEyuQuel3+4P3Z5rtxOQzja3Wr0ganue5Vqtb61WMErswHfO3AfNh1ar77zKkxoPzzarjXqDL2t+0DSDqjKvg+DO7Z9US3VVoGw1Ur5y1FAlGHuqrWK8X6imaqgQtJdqDFqEkct5R03UBmRH4HLA0QZ1gO8+fr0w1CF+C2ZMaRtaPHwiSFrqC8PTxbhHqn6KfivHu0zHmNhi4xWeHYPpg5qoc1DL5FLO9eU6ePslq06whm+4WhcrCUkRP9iFNffw9PA7wQrl+wqcDkZdSEUY2aB5oGqK6Ijw1J4X35wzEm3yORiJ1avsXmW1zAd4D4DVxjimpWKrpZ6auAyp2aGtwuMxpssRf8UKtX2rsHrTNbiMu16F8B6CMlCvMSoir9KZz67lXIlBnizQJfwJOYbgjkEJmPku/OeCo5irNjBFR5uR7HMtITOobpB/4HyazSHiuUV7YkbNYr651BOaysrsTu308OwQO4L8GHPn1CO+qMEeh9gRszf1Yo3cEX5d8pdNG+0Zep2VKfGsYU2SXTXgBphxp1jaE9qfqUXaujFolnlvqR+p30Hca/SK5HUNsgEt1FkcU9fQxH4XfUX84uNbqOLZlFKjDp92nDNbBqBl+jSS6PsSq4hZP55BHePp0WshUWrULv655NjnmiSiI+q+xLhLPSJtodfV1Y6xY1LQKr5x2Z3mtVrMP6ld3RlntYrckJVqmazcpaaGemRWLBEQu/Nes0031D055uomM1Z1aH/WNSRHi/aJVBafYqy6zMZzShVxxYuL17loLduskxpjKb7vg2OXUj1Dj1mbodG0kau/PVMlAeNh8+mbnND1LtouZ2bGoPvGUx74dJ3LniQzr4DZZk616L38/CrEJ0Y+3Q2k449JXS13zMoqykYYjaczq+Vdjhz6R48k4hovpK9DruXz6cfKzayL/XbwhnOIkrGJQVoXvcvcWLzyQ1VFL6giq4r+lEgLJeLpJMT4/pTzPnCFd0AtQ1CkwsfT+UlpDPZBmRiP9TiX1mFmg8vO1yVny2Se5tAVlcxwSFWv1ip5uShvNH21bHea70VZTV8tK5QL5qZr+kFMrNMSuYRxEAk/l6WHJVIB9wRN1zb/vIZ9Q+61I5MXkbHwpETSZ4/RqwpMVT4rtS9hF4yMT0Tmlzfw4AX734QnoOv6MZNNruXNTO7qjXyayV9e07OZpH9N/2aSr0skHe4N7pQmMs9La0q4Dkq5dD6V7QkOdpP0/KKrrcbvDrwoe3bXdCbL7DeSFXp33GUPqK3Vn+f3HankZmkdj9h9O+xDETV3V9TO0VtDyyI4Mnt5upunp/A41+ddasxO6GOepi7NjpCeq7boSe2L79FTnuGsIuM99RXOSU18Py3sHuuiyn1lZPpOHncbyDvo+/vcH66PG5pbxVbh7lHU8QQa0s/E3NIybs+cpZchXUfvYmQdve7cmX5Wkz61nYNL32SuShCzfFiMtzq+i5H1Pw4BT9LZfeZtIOfvauvhZTmz/GZddt9ffgsN8ezh22NnjA3vPrPOY4d0UKv6pqxPIXtzd37J321UheTu+N7dGX1F/NlbZJ+3wBE4Mh59G0gorWl5xGX30ZAyiTl1xby7ZTfUeuHGqfFmbYtzeeUbvF3ToRzTdTfONqvN2X+x5gfH2/Xm1/XG80fVx9+Zf7huqc/UPfUAftpRj1GZB/CrrX5Xf6t/1L+VQeW3yh+VPzXrzRtGpqIKr8pf/wPVt99m</latexit>

Yi (0)

Means every ITE is nonnegative, so ITE lower bound comes up from a – b to 0
Intuitively, the ATE lower bound should raise up to 0 too<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

E[Y (1)

Y (0)]

0

Proof:<latexit 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sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">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</latexit><latexit sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">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</latexit><latexit sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">AAAVkHicrVhZb9tGEN64V+Jetgs/9WUbpUGCyKrkpnBQQGgK20ERNIBTOHHa0HAlcmWz4mWSsusI+pV96r/pzLdLkdRFObUEi8vZmW9m59pddyPPTdJm899bKx98+NHHn9y+s/rpZ59/8eXa+sbrJBzEtnplh14Yv+l2EuW5gXqVuqmn3kSx6vhdTx11+7s8f3Sh4sQNg8P0KlLHfuc0cHuu3UmJdLJ+6x9r/60VDlI79NWD1kO5JcdvzYfH8n5bWpErrbos8EnLdx1ppaQo9VWQyrZsHctH8kGLxSP34QQ/406LNI+lFYTBwO+qWFrWqqTPfet80HE0yiRIcxYI6d2aq3eO0kzVqTq/gcX9/5Xd7LLo5WSt1mw08ZHTg5YZ1IT5HITrd34TlnBEKGwxEL5QIhApjT3REQl934qWaIqIaMdiSLSYRi7mlRiJVZIdEJcijg5R+/R7Sm9vDTWgd8ZMIG2TFo/+YpKU4lvD49C4B6p+sn5Z4J2nYwhstvGKnl2D6RM1FWdErZLLOJeX69LXr1h1Smt4gtW6tJIIFPaDXVpzj54evae0Qv69Ik5FI4ekYhrZRPOIqimsI6an9jz75gyR6IBP0YitXmT3Iqt5PqRvn7A6NE5gKdsqxTMTlwCaFWxlHg8xnY/4N61Q27cIqzdeg4u461Uw7yFR+uIdjcrIi3QWs2s+V2qQRzN0MX8KjoC4E6KEyHyX/OcSRzlXbcJkHR1E8hRriZBBDYP8HPNZNkcUzy3YkyBqEvnmQk9kKiu3O7PTo2cX2DHJD2nuDHrYF3WyRwE7RvZmXqyDO6a3S7zZsNGeoDdQmRzPOq2Js6tOuCHNuGMs7Qntz8wibd2QaNJ8t8Sv0K8o7nV4hfO6TrIhLNRZnEBXYGLfpr7CfvHpl6ns2YxShw4fdpwhW/pEy/VpJNb3Ha0iQf14BnVITw9ei4BSh3b2zyXGPtbEER1A9yWNHehhaUm9riF2jB2jklb2jYvuNK1VIv+4dnVnnNTKcgEqVZqsbENTUzw2K+YIsN1Fr9mmG+qenGB1owmrurA/7xqco2X7WCqPTzlWDrLxDFJlXPbi7HXOWss26qSOWLLvT4mjDameoSeozchoWi3U366pkhDxsPH0TU7oemdtlxMzQ6L7xlMe8ek65z2JZ84Js4OcsuC94vwixH0jn+0G3PGHoC6We43KKsvGNBqOZxbLuxgp+EePOOIaL4KvI6zlm/GfLMwsi30zeMEUImdsapCWRXeQG7NXfihq1AtqlFVlf3KkmRLjdBLR+N6Y8x7hMm8fWgKicIUPx/OjyhjsEWVkPNbDXFaHuQ0uOp8DTstknubQFZVOcHBVL9bKeTkrbzR9sawzzveyrKYvlmXKBXLTNf0gAdabCrkUcWAJv5ClhxVSIfYETdc2/76EfQH22oHJi9hYeFQh6aPH6FWFpipfVNqXogvGxics88d7ePAC/W+EE9B1/ZjLptfyZi539V4+zeUvr+nZXNK/pn9zyXcVkgp7gzumsczLyppiroNKLp1PVXuCot0kO7/oaqvjt0te5D3bMZ1Jmv2Gs0Lvjm30gPpS/Xl63+FKblXW8QDdt4s+FEOzs6B2Xt0YWh7BgdnLs908O4UnhT7vQmN+Qh/iNHVpdoTsXLUFT2pf/EI95QWdVXi8K76nc1KLfp+Vdo9lUfm+MjB9p4i7Tcg71Pf3sD9cHzcyt4qt0t2jrGOfNGR/I3NLy7k9c5aeh3QdvbORdfScqTP9pCZ9ajsjLn2TuapAzPNhNt7i+M5G1v9xCHGSzu8zN4FcvKsth5fnzPybddV9f/4tNKJnj349dMbE8O4h6zx0SEW1qm/K+hSyO3Xn5/zdpqrg3B3e3ZjQV8afvEWe4hY4II6cR98GUkhrWhFx3n00gkxqTl0J7m75DbVRunFqvEnbkkJe+QavbTqUMl139WSt1pr8L9b04PV2o/VDo/nyce3pE/Mfrtvia3FXPCA/7YinVJkH5Fd75ceVP1fclb82NzafbP60+bNmXbllZL4Spc/m8/8AIrJeWw==</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

⇡ E[Y | T = 1] + (1 ⇡) E[Y(0) =| T 0]= 1]⇡ E[Y | T = 1] (1 ⇡) E[YE[Y| T(1) | T = 0]
| T E[Y= 0]<latexit 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| T = 0] = E[Y (0)| T |=
E[Y (1)
T 0]= 0] = E(1) | T = 1] =E[Y (0)| T |=
E[Y (1)
T 1]= 1] ==0
Brady Neal

Bounds: Monotone Treatment Response

14 / 48

E

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">AAAT6XicrVjbbttGEN2kt8S9OdFjX9goBRJAViU3hQu0BgLYCYqiAZzCtzYKDImiZEKkyJCUXUfQRxR9Kfrar+hr+xX9m545uxRJ3SgHkSBxNTtzZnZuu6tO6Llx0mj8d+PmO+++9/4Ht25vfPjRx598unnn7nEcjCLbObIDL4hOO+3Y8dyhc5S4ieechpHT9juec9IZ7Mn8yYUTxW4wPEyuQuel3+4P3Z5rtxOQzja3Wr0ganue5Vqtb61WMErswHfO3AfNh1ar77zKkxoPzzarjXqDL2t+0DSDqjKvg+DO7Z9US3VVoGw1Ur5y1FAlGHuqrWK8X6imaqgQtJdqDFqEkct5R03UBmRH4HLA0QZ1gO8+fr0w1CF+C2ZMaRtaPHwiSFrqC8PTxbhHqn6KfivHu0zHmNhi4xWeHYPpg5qoc1DL5FLO9eU6ePslq06whm+4WhcrCUkRP9iFNffw9PA7wQrl+wqcDkZdSEUY2aB5oGqK6Ijw1J4X35wzEm3yORiJ1avsXmW1zAd4D4DVxjimpWKrpZ6auAyp2aGtwuMxpssRf8UKtX2rsHrTNbiMu16F8B6CMlCvMSoir9KZz67lXIlBnizQJfwJOYbgjkEJmPku/OeCo5irNjBFR5uR7HMtITOobpB/4HyazSHiuUV7YkbNYr651BOaysrsTu308OwQO4L8GHPn1CO+qMEeh9gRszf1Yo3cEX5d8pdNG+0Zep2VKfGsYU2SXTXgBphxp1jaE9qfqUXaujFolnlvqR+p30Hca/SK5HUNsgEt1FkcU9fQxH4XfUX84uNbqOLZlFKjDp92nDNbBqBl+jSS6PsSq4hZP55BHePp0WshUWrULv655NjnmiSiI+q+xLhLPSJtodfV1Y6xY1LQKr5x2Z3mtVrMP6ld3RlntYrckJVqmazcpaaGemRWLBEQu/Nes0031D055uomM1Z1aH/WNSRHi/aJVBafYqy6zMZzShVxxYuL17loLduskxpjKb7vg2OXUj1Dj1mbodG0kau/PVMlAeNh8+mbnND1LtouZ2bGoPvGUx74dJ3LniQzr4DZZk616L38/CrEJ0Y+3Q2k449JXS13zMoqykYYjaczq+Vdjhz6R48k4hovpK9DruXz6cfKzayL/XbwhnOIkrGJQVoXvcvcWLzyQ1VFL6giq4r+lEgLJeLpJMT4/pTzPnCFd0AtQ1CkwsfT+UlpDPZBmRiP9TiX1mFmg8vO1yVny2Se5tAVlcxwSFWv1ip5uShvNH21bHea70VZTV8tK5QL5qZr+kFMrNMSuYRxEAk/l6WHJVIB9wRN1zb/vIZ9Q+61I5MXkbHwpETSZ4/RqwpMVT4rtS9hF4yMT0Tmlzfw4AX734QnoOv6MZNNruXNTO7qjXyayV9e07OZpH9N/2aSr0skHe4N7pQmMs9La0q4Dkq5dD6V7QkOdpP0/KKrrcbvDrwoe3bXdCbL7DeSFXp33GUPqK3Vn+f3HankZmkdj9h9O+xDETV3V9TO0VtDyyI4Mnt5upunp/A41+ddasxO6GOepi7NjpCeq7boSe2L79FTnuGsIuM99RXOSU18Py3sHuuiyn1lZPpOHncbyDvo+/vcH66PG5pbxVbh7lHU8QQa0s/E3NIybs+cpZchXUfvYmQdve7cmX5Wkz61nYNL32SuShCzfFiMtzq+i5H1Pw4BT9LZfeZtIOfvauvhZTmz/GZddt9ffgsN8ezh22NnjA3vPrPOY4d0UKv6pqxPIXtzd37J321UheTu+N7dGX1F/NlbZJ+3wBE4Mh59G0gorWl5xGX30ZAyiTl1xby7ZTfUeuHGqfFmbYtzeeUbvF3ToRzTdTfONqvN2X+x5gfH2/Xm1/XG80fVx9+Zf7huqc/UPfUAftpRj1GZB/CrrX5Xf6t/1L+VQeW3yh+VPzXrzRtGpqIKr8pf/wPVt99m</latexit>

Yi (0)

Means every ITE is nonnegative, so ITE lower bound comes up from a – b to 0
Intuitively, the ATE lower bound should raise up to 0 too<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

E[Y (1)

Y (0)]

0

Proof:<latexit 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sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">AAAVkHicrVhZb9tGEN64V+Jetgs/9WUbpUGCyKrkpnBQQGgK20ERNIBTOHHa0HAlcmWz4mWSsusI+pV96r/pzLdLkdRFObUEi8vZmW9m59pddyPPTdJm899bKx98+NHHn9y+s/rpZ59/8eXa+sbrJBzEtnplh14Yv+l2EuW5gXqVuqmn3kSx6vhdTx11+7s8f3Sh4sQNg8P0KlLHfuc0cHuu3UmJdLJ+6x9r/60VDlI79NWD1kO5JcdvzYfH8n5bWpErrbos8EnLdx1ppaQo9VWQyrZsHctH8kGLxSP34QQ/406LNI+lFYTBwO+qWFrWqqTPfet80HE0yiRIcxYI6d2aq3eO0kzVqTq/gcX9/5Xd7LLo5WSt1mw08ZHTg5YZ1IT5HITrd34TlnBEKGwxEL5QIhApjT3REQl934qWaIqIaMdiSLSYRi7mlRiJVZIdEJcijg5R+/R7Sm9vDTWgd8ZMIG2TFo/+YpKU4lvD49C4B6p+sn5Z4J2nYwhstvGKnl2D6RM1FWdErZLLOJeX69LXr1h1Smt4gtW6tJIIFPaDXVpzj54evae0Qv69Ik5FI4ekYhrZRPOIqimsI6an9jz75gyR6IBP0YitXmT3Iqt5PqRvn7A6NE5gKdsqxTMTlwCaFWxlHg8xnY/4N61Q27cIqzdeg4u461Uw7yFR+uIdjcrIi3QWs2s+V2qQRzN0MX8KjoC4E6KEyHyX/OcSRzlXbcJkHR1E8hRriZBBDYP8HPNZNkcUzy3YkyBqEvnmQk9kKiu3O7PTo2cX2DHJD2nuDHrYF3WyRwE7RvZmXqyDO6a3S7zZsNGeoDdQmRzPOq2Js6tOuCHNuGMs7Qntz8wibd2QaNJ8t8Sv0K8o7nV4hfO6TrIhLNRZnEBXYGLfpr7CfvHpl6ns2YxShw4fdpwhW/pEy/VpJNb3Ha0iQf14BnVITw9ei4BSh3b2zyXGPtbEER1A9yWNHehhaUm9riF2jB2jklb2jYvuNK1VIv+4dnVnnNTKcgEqVZqsbENTUzw2K+YIsN1Fr9mmG+qenGB1owmrurA/7xqco2X7WCqPTzlWDrLxDFJlXPbi7HXOWss26qSOWLLvT4mjDameoSeozchoWi3U366pkhDxsPH0TU7oemdtlxMzQ6L7xlMe8ek65z2JZ84Js4OcsuC94vwixH0jn+0G3PGHoC6We43KKsvGNBqOZxbLuxgp+EePOOIaL4KvI6zlm/GfLMwsi30zeMEUImdsapCWRXeQG7NXfihq1AtqlFVlf3KkmRLjdBLR+N6Y8x7hMm8fWgKicIUPx/OjyhjsEWVkPNbDXFaHuQ0uOp8DTstknubQFZVOcHBVL9bKeTkrbzR9sawzzveyrKYvlmXKBXLTNf0gAdabCrkUcWAJv5ClhxVSIfYETdc2/76EfQH22oHJi9hYeFQh6aPH6FWFpipfVNqXogvGxics88d7ePAC/W+EE9B1/ZjLptfyZi539V4+zeUvr+nZXNK/pn9zyXcVkgp7gzumsczLyppiroNKLp1PVXuCot0kO7/oaqvjt0te5D3bMZ1Jmv2Gs0Lvjm30gPpS/Xl63+FKblXW8QDdt4s+FEOzs6B2Xt0YWh7BgdnLs908O4UnhT7vQmN+Qh/iNHVpdoTsXLUFT2pf/EI95QWdVXi8K76nc1KLfp+Vdo9lUfm+MjB9p4i7Tcg71Pf3sD9cHzcyt4qt0t2jrGOfNGR/I3NLy7k9c5aeh3QdvbORdfScqTP9pCZ9ajsjLn2TuapAzPNhNt7i+M5G1v9xCHGSzu8zN4FcvKsth5fnzPybddV9f/4tNKJnj349dMbE8O4h6zx0SEW1qm/K+hSyO3Xn5/zdpqrg3B3e3ZjQV8afvEWe4hY4II6cR98GUkhrWhFx3n00gkxqTl0J7m75DbVRunFqvEnbkkJe+QavbTqUMl139WSt1pr8L9b04PV2o/VDo/nyce3pE/Mfrtvia3FXPCA/7YinVJkH5Fd75ceVP1fclb82NzafbP60+bNmXbllZL4Spc/m8/8AIrJeWw==</latexit><latexit 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E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

⇡ E[Y | T = 1] + (1 ⇡) E[Y(0) =| T 0]= 1]⇡ E[Y | T = 1] (1 ⇡) E[YE[Y| T(1) | T = 0]
| T E[Y= 0]<latexit 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| T = 0] = E[Y (0)| T |=
E[Y (1)
T 0]= 0] = E(1) | T = 1] =E[Y (0)| T |=
E[Y (1)
T 1]= 1] ==0
Brady Neal

Bounds: Monotone Treatment Response

14 / 48

E

Nonnegative Monotone Treatment Response (MTR)
Assume treatment always helps. Mathematically, 8i Yi (1)<latexit sha1_base64="k7MAJGIU+MLxQVWTVSGao7w9rMU=">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</latexit>

Yi (0)

Means every ITE is nonnegative, so ITE lower bound comes up from a – b to 0
Intuitively, the ATE lower bound should raise up to 0 too<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

E[Y (1)

Y (0)]

0

Proof:<latexit 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sha1_base64="4hprTbdz8WVJZ1uOc2M2KYZHpfM=">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</latexit><latexit 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E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

⇡ E[Y | T = 1] + (1 ⇡) E[Y(0) =| T 0]= 1]⇡ E[Y | T = 1] (1 ⇡) E[YE[Y| T(1) | T = 0]
| T E[Y= 0]<latexit 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| T = 0] = E[Y (0)| T |=
E[Y (1)
T 0]= 0] = E(1) | T = 1] =E[Y (0)| T |=
E[Y (1)
T 1]= 1] ==0
Brady Neal

Bounds: Monotone Treatment Response

14 / 48

E

Running Example: Nonnegative MTR
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

Brady NealE[Y | T = 1] = .9

<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

Bounds: Monotone Treatment Response

15 / 48

Running Example: Nonnegative MTR
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

No-assumptions bound:

Brady Neal<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit><latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

0.17  E[Y (1)Y (0)]  0.83

Bounds: Monotone Treatment Response

15 / 48

Running Example: Nonnegative MTR
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">AAAT4XicrVjLbttGFJ2kr8R9OfGyGzZOgC5kRXJTOF0ICGA7KIoGcAq/2sgwJIqyCPNVkrLqCPqAopui235Ft+2X9G967pmhKOpFObAFk6M79557575mRu3Ic5O0Vvvvzt333v/gw4/u3V/7+JNPP/t8/cHD4yTsx7ZzZIdeGJ+2W4njuYFzlLqp55xGsdPy255z0r7clfmTKydO3DA4TK8j58xvXQRu17VbKUjn60+a+2+aYT+1Q9+xmr7bsZop5FPfCVKrYdXP8Kh+e76+WavW+GfNDupmsKnM30H44P6Pqqk6KlS26itfOSpQKcaeaqkEnzeqrmoqAu1MDUGLMXI576iRWoNsH1wOOFqgXuJ5gW9vDDXAd8FMKG1Di4f/GJKWemJ4Ohh3SdVv0W9N8C7SMSS22HiNd9tg+qCmqgdqmVzGubpcGx+/ZNUp1vCcq3WxkogU8YNdWHMXbw/fU6xQntfgdDDqQCrGyAbNA1VTREeMt/a8+KbHSLTI52AkVi+ze5nVMh/icwmsFsYJLRVbLfXSxCWgZoe2Co/HmC5G/BUr1PYtw+qO1+Ay7noVwnsIyqV6i1EReZnOyexazJUa5NEcXcKfkiMAdwJKyMx34T8XHMVctYEpOlqM5AXXEjGDqgb5e85n2Rwhnlu0J2HULOabSz2Rqazc7sxOD+82sWPIDzHXox7xRQX2OMSOmb2ZFyvkjvFtwG82bbSn6FVWpsSzgjVJdlWAG2LGHWNpT2h/ZhZp64agWeazpX6gfgdxr9ArktcVyIa0UGdxQl2BiX0DfUX84uMpVPFsRqlQh087esyWS9ByfRpJ9D3FKhLWj2dQh3h79FpElAq1i38GHPtck0S0T90DjDvUI9IWel1V7Rg7RgWt4huX3WlWq8X8k9rVnXFaq8gFrFTLZGWDmmrqmVmxREDsnvSabbqh7skJVzeasqpN+/OuITlatE+k8vgUY9VhNvYoVcQVL85f57y1bLNOKoyl+P4CHA1KdQ09YW1GRtPaRP3tmioJGQ+bb9/khK530TaYmhmC7htPeeDTdS57ksz8AswWc6pJ703OL0PcN/LZbiAdf0jqcrljVlZRNsZoOJ5ZLu9y5NA/eiQR13gRfR1xLV+O/62JmVWxbwcvmEGUjE0N0qroHebG/JUfqk30gk1kVdGfEmmhxDydRBg/HnM+Bq7wXlJLAIpU+HA8PyqNwR4oI+OxLueyOsxtcNn5OuRsmszTHLqi0ikOqerlWiUv5+WNpi+X7YzzvSir6ctlhXLF3HRNP0iIdVoilzIOIuFPZOlhiVTIPUHTtc0/rWBfwL22b/IiNhaelEj67DF6VaGpylel9qXsgrHxicj8/A4evGL/G/EEdFM/5rLpjbyZy12/k09z+cENPZtL+jf0by75tkTS4d7gjmki87q0poTroJRL51PZnuBgN8nOL7raKny24UXZszumM1lmv5Gs0Ltjgz2gslJ/nt13pJLrpXXcZ/dtsw/F1NxZUjtHt4aWR7Bv9vJsN89O4clEn3epMT+hD3maGpgdITtXbdGT2hffoae8wllFxrvqa5yT6ni+LOweq6LKfaVv+s4k7jaQd9D397g/3Bw3MreKrcLdo6hjHxqy/5G5peXcnjlLL0K6id75yDp6nZkz/bQmfWrrgUvfZK5LEPN8mI+3PL7zkfUvDiFP0vl95jaQJ+9qq+HlObP4Zl123198C43w7uLpsTMmhnePWeexQzqoVX1T1qeQ3Zk7v+TvNqpCcnf46OGUviL+9C3ygrfAPjhyHn0bSCmtaZOIi+6jEWVSc+pKeHfLb6jVwo1T403blkzklW/wGqZDOabrrp2vb9anf8WaHRxvV+vfVGuvn22+eG5+4bqnvlCP1Ffw0456gco8gF9t9bv6W/2j/t2wN37b+GPjT816946R2VCFv42//geLj9wD</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

No-assumptions bound:<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">AAAT7nicrVjbbttGEN0kvSTuzYke88JGKZAAkio5KewXAwFsB0XRAE7hW2sZhkStLEK8haSsOoR+o+hL0dd+RV/bX+jfdObsUiR1IaXAEiSuZmfOzM5td9X1bSuMms3/7ty999HHn3x6/8HGZ59/8eVXmw8fnYTeKDDlsenZXnDW7YTStlx5HFmRLc/8QHacri1Pu8M9nj+9lkFoee5RdOPLC6dz5Vp9y+xERLrc3Ko3G61to23Ld0b74LztjSLTc+Sz1nOjbkx/NZ9fKJZmY+fF5Wa12WjiZcwPWnpQFfp16D188JNoi57whClGwhFSuCKisS06IqT3uWiJpvCJdiFiogU0sjAvxURskOyIuCRxdIg6pO8r+nWuqS79ZswQ0iZpsekTkKQhvtE8PRr3QVVP1m9keJfpiIHNNt7Qs6sxHaJGYkDUMrmEc3W5Lr2dklVHtIYdrNailfigsB/M3Jr79LTpd0Qr5O8b4pQ06pFUQCOTaDZRFYV1BPRUnmffDBCJDvgkjdjqIruLrOZ5j95DwurQOISlbKshXuu4uNAsYSvz2IjpcsRfaYXKviKs/nQNFuKuVsG8R0QZivc0yiMX6cxm13KuSCNPFuhi/ggcLnGHRPGQ+Rb5zyKOfK6ahMk6OojkFdbiI4MaGvkHzCfZ7FM867AnRNQM5JsFPb6urNTuxE6bnl1gByQf09wAetgXNbJHAjtA9iZerIE7oF9j/DJhozlDb6AyOZ41WhNnV41wPZqxpljKE8qfiUXKuphohn7XxY/QLynuNXiF87pGsh4sVFkcQperY79LfYX94tA3U9mzCaUGHQ7sGCBbhkRL9Skk1vctrSJE/dgaNaanDa/5QKlBO/tnjLGDNXFER9A9pnEPeljaoF7XENvajklOK/vGQnea12og/7h2VWec1cpyLirV0Fm5C01N8VKvmCPAdme9ZupuqHpyiNVNZqzqwv60a3CO5u1jqTQ++Vj1kI0DSOVx2YuL17loLVuokxpiyb6/Io5dSPU1PURt+lrTRqb+9nSVeIiHiaejc0LVO2sbz8zERHe0p2ziU3XOexLPvCPMDnKqDe9l54sQD7R8shtwx49BLZY7QWXlZQMaxdOZYnkLIwn/qBFHXOH58LWPtXw9/RiZmVWxbwfPnUPkjI000qroPeTG4pUfiSr1giplVd6fHGmmBDid+DR+OuV8SrjMO4QWlyhc4fF0flIag32iTLTH+phL6jC1wULn64GzrTNPcaiKimY4uKqLtXJeLsobRS+W7U3zPS+r6MWyTLlGblq6H4TAOiuRixAHlnAyWXpUIuVhT1B0ZfPPK9jnYq8d6bwItIWnJZIOeoxalaer8k2pfRG6YKB9wjK/fIAHr9H/JjgBrevHVDZay5up3M0H+TSVH6/p2VTSWdO/qeT7EkmJvcGa0ljmbWlNMddhKZfKp7I9QdJukpxfVLXV8N0lL/Ke3dOdydD7DWeF2h130QNqK/Xn+X2HK7lVWscjdN8u+lAAzb2C2jm+NbQ0giO9lye7eXIKDzN93oLG9IQe4zQ11jtCcq6qw5PKF99TT3lDZxUe74kXdE5q0ffr3O6xKirfV0a672Rxtwh5m/r+PvaH9XF9fauo5+4eeR0HpCH5TPQtLeW29Vl6GdI6ehcjq+j15s70s5rUqW1AXOomc1OCmObDYrzi+C5GVv84eDhJp/eZ20DO3tVWw0tzZvnNuuy+v/wW6tOzT982OmOoefeRdTY6pKRaVTdldQrZm7vzc/5uUVVw7sZPHs3oy+PP3iKvcAscEUfKo24DEaQVLYu47D7qQybSp64Qd7f0htrI3TgV3qxtYSavHI23qzuU1F1343Kz2pr9F2t+cLLVaH3XaL59WX21o//hui8eiyfiGflpW7yiyjwkv5rid/G3+Ef8W/Erv1X+qPypWO/e0TIVkXtV/vofN6/fLQ==</latexit><latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

0.17  E[Y (1)Nonnegative MTR lower bound: E[Y (1)Y (0)]  0.83

<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

Brady Neal

Y (0)]

0

Bounds: Monotone Treatment Response

15 / 48

Running Example: Nonnegative MTR
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

No-assumptions bound:<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit><latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

0.17  E[Y (1)Nonnegative MTR lower bound: E[Y (1)Y (0)]  0.83

<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

Y (0)]

0

Combining MTR lower bound with no-assumptions upper bound:
0  E[Y (1)<latexit sha1_base64="X3OY3o8+SIhIpXLI1dE7gFVTT00=">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</latexit>

Brady Neal

Y (0)]  0.83

Bounds: Monotone Treatment Response

15 / 48

Nonpositive Monotone Treatment Response<latexit sha1_base64="qMa/GNp1MgL3a3mxtYiyioBmP44=">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</latexit>

Brady Neal

8i Yi (1)  Yi (0)Bounds: Monotone Treatment Response

16 / 48

Nonpositive Monotone Treatment Response<latexit sha1_base64="qMa/GNp1MgL3a3mxtYiyioBmP44=">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</latexit><latexit sha1_base64="YGYZVY8rfAiR1ujGVWT51yYAS70=">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</latexit>

Brady Neal

8i Yi (1)  Yi (0)E[Y (1)

Y (0)]  0Bounds: Monotone Treatment Response

16 / 48

Nonpositive Monotone Treatment Response<latexit sha1_base64="qMa/GNp1MgL3a3mxtYiyioBmP44=">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</latexit><latexit sha1_base64="YGYZVY8rfAiR1ujGVWT51yYAS70=">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</latexit>

8i Yi (1)  Yi (0)E[Y (1)

Y (0)]  0

Running example
No-assumptions bound:

Brady Neal<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">AAAT7nicrVjbbttGEN0kvSTuzYke88JGKZAAkio5KewXAwFsB0XRAE7hW2sZhkStLEK8haSsOoR+o+hL0dd+RV/bX+jfdObsUiR1IaXAEiSuZmfOzM5td9X1bSuMms3/7ty999HHn3x6/8HGZ59/8eVXmw8fnYTeKDDlsenZXnDW7YTStlx5HFmRLc/8QHacri1Pu8M9nj+9lkFoee5RdOPLC6dz5Vp9y+xERLrc3Ko3G61to23Ld0b74LztjSLTc+Sz1nOjbkx/NZ9fKJZmY+fF5Wa12WjiZcwPWnpQFfp16D188JNoi57whClGwhFSuCKisS06IqT3uWiJpvCJdiFiogU0sjAvxURskOyIuCRxdIg6pO8r+nWuqS79ZswQ0iZpsekTkKQhvtE8PRr3QVVP1m9keJfpiIHNNt7Qs6sxHaJGYkDUMrmEc3W5Lr2dklVHtIYdrNailfigsB/M3Jr79LTpd0Qr5O8b4pQ06pFUQCOTaDZRFYV1BPRUnmffDBCJDvgkjdjqIruLrOZ5j95DwurQOISlbKshXuu4uNAsYSvz2IjpcsRfaYXKviKs/nQNFuKuVsG8R0QZivc0yiMX6cxm13KuSCNPFuhi/ggcLnGHRPGQ+Rb5zyKOfK6ahMk6OojkFdbiI4MaGvkHzCfZ7FM867AnRNQM5JsFPb6urNTuxE6bnl1gByQf09wAetgXNbJHAjtA9iZerIE7oF9j/DJhozlDb6AyOZ41WhNnV41wPZqxpljKE8qfiUXKuphohn7XxY/QLynuNXiF87pGsh4sVFkcQperY79LfYX94tA3U9mzCaUGHQ7sGCBbhkRL9Skk1vctrSJE/dgaNaanDa/5QKlBO/tnjLGDNXFER9A9pnEPeljaoF7XENvajklOK/vGQnea12og/7h2VWec1cpyLirV0Fm5C01N8VKvmCPAdme9ZupuqHpyiNVNZqzqwv60a3CO5u1jqTQ++Vj1kI0DSOVx2YuL17loLVuokxpiyb6/Io5dSPU1PURt+lrTRqb+9nSVeIiHiaejc0LVO2sbz8zERHe0p2ziU3XOexLPvCPMDnKqDe9l54sQD7R8shtwx49BLZY7QWXlZQMaxdOZYnkLIwn/qBFHXOH58LWPtXw9/RiZmVWxbwfPnUPkjI000qroPeTG4pUfiSr1giplVd6fHGmmBDid+DR+OuV8SrjMO4QWlyhc4fF0flIag32iTLTH+phL6jC1wULn64GzrTNPcaiKimY4uKqLtXJeLsobRS+W7U3zPS+r6MWyTLlGblq6H4TAOiuRixAHlnAyWXpUIuVhT1B0ZfPPK9jnYq8d6bwItIWnJZIOeoxalaer8k2pfRG6YKB9wjK/fIAHr9H/JjgBrevHVDZay5up3M0H+TSVH6/p2VTSWdO/qeT7EkmJvcGa0ljmbWlNMddhKZfKp7I9QdJukpxfVLXV8N0lL/Ke3dOdydD7DWeF2h130QNqK/Xn+X2HK7lVWscjdN8u+lAAzb2C2jm+NbQ0giO9lye7eXIKDzN93oLG9IQe4zQ11jtCcq6qw5PKF99TT3lDZxUe74kXdE5q0ffr3O6xKirfV0a672Rxtwh5m/r+PvaH9XF9fauo5+4eeR0HpCH5TPQtLeW29Vl6GdI6ehcjq+j15s70s5rUqW1AXOomc1OCmObDYrzi+C5GVv84eDhJp/eZ20DO3tVWw0tzZvnNuuy+v/wW6tOzT982OmOoefeRdTY6pKRaVTdldQrZm7vzc/5uUVVw7sZPHs3oy+PP3iKvcAscEUfKo24DEaQVLYu47D7qQybSp64Qd7f0htrI3TgV3qxtYSavHI23qzuU1F1343Kz2pr9F2t+cLLVaH3XaL59WX21o//hui8eiyfiGflpW7yiyjwkv5rid/G3+Ef8W/Erv1X+qPypWO/e0TIVkXtV/vofN6/fLQ==</latexit>

0.17  E[Y (1)Y (0)]  0.83

Bounds: Monotone Treatment Response

16 / 48

Nonpositive Monotone Treatment Response<latexit sha1_base64="qMa/GNp1MgL3a3mxtYiyioBmP44=">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</latexit><latexit sha1_base64="YGYZVY8rfAiR1ujGVWT51yYAS70=">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</latexit>

8i Yi (1)  Yi (0)E[Y (1)

Y (0)]  0

Running example
No-assumptions bound:<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit>

0.17  E[Y (1)Y (0)]  0.83

Combining nonpositive MTR upper bound with no-assumptions lower bound:<latexit sha1_base64="vrnKiofqpzN5cAjGt9p7eOnwHAg=">AAAT63icrVjbbttGEN2kt8S9OdFjX9goBRKAUiQ3hfNiIIDtoCgawCl8ay3DkKiVRIi3kJRVR9BXFH0p+tqv6Gv7Ef2bzpxdiqQupBRYgsTV7MyZ2bntrjqBY0dxo/HfnbsffPjRx5/cu7/16Weff/Hl9oOHp5E/Ci15YvmOH5532pF0bE+exHbsyPMglG2348izznCf58+uZRjZvncc3wTy0m33PbtnW+2YSFfbz2qNenPXaDnyrdE6vGj5o9jyXfmk+dSoGbNfjaeXiqVxtV1t1Bt4GYuDph5UhX4d+Q/u/yRaoit8YYmRcIUUnohp7Ii2iOh9IZqiIQKiXYoJ0UIa2ZiXYiq2SHZEXJI42kQd0neffl1oqke/GTOCtEVaHPqEJGmIbzRPl8Y9UNWT9RsZ3lU6JsBmG2/o2dGYLlFjMSBqmVzCub5ch95uyapjWsMLrNamlQSgsB+s3Jp79HTod0wr5O8b4pQ06pJUSCOLaA5RFYV1hPRUnmffDBCJNvgkjdjqIruLrOZ5n95DwmrTOIKlbKshXum4eNAsYSvzOIjpasRfaYXKviKs3mwNNuKuVsG8x0QZinc0yiMX6cxm12quWCNPl+hi/hgcHnFHRPGR+Tb5zyaOfK5ahMk62ohkH2sJkEF1jfwD5pNsDiieNdgTIWoG8s2GnkBXVmp3YqdDzw6wQ5Kf0NwAetgXJtkjgR0iexMvmuAO6dcYvyzYaM3R66hMjqdJa+LsMgnXpxl7hqU8ofyZWKSsmxDN0O+a+BH6JcXdhFc4r02S9WGhyuIIujwd+z3qK+wXl76Zyp5NKCZ0uLBjgGwZEi3Vp5BY3zNaRYT6cTTqhJ4OvBYAxYR29s8YYxdr4oiOoHtM4y70sLRBva4udrUd05xW9o2N7rSo1UD+ce2qzjivleU8VKqhs3IPmhriuV4xR4DtznrN0t1Q9eQIq5vOWdWB/WnX4BzN28dSaXzyseoiGweQyuOyF5evc9ladlAnJmLJvu8Txx6kepoeoTYDrWkrU3/7ukp8xMPC09U5oeqdtY3nZiZEd7WnHOJTdc57Es+8Jcw2cqoF72XnixAPtXyyG3DHn4BaLHeKysrLhjSazGaK5W2MJPyjRhxxhRfA1wHW8vXsY2Rm1sW+HTxvAZEzNtZI66J3kRvLV34sqtQLqpRVeX9ypJkS4nQS0PjxjPMx4TLvEFo8onCFT2bz09IYHBBlqj3Ww1xSh6kNNjpfF5wtnXmKQ1VUPMfBVV2slfNyWd4oerFsd5bveVlFL5ZlyjVy09b9IALWeYlcjDiwhJvJ0uMSKR97gqIrm39ewz4Pe+1I50WoLTwrkXTRY9SqfF2Vr0vti9EFQ+0TlvnlPTx4jf43xQloUz+msvFG3kzlbt7Lp6n8eEPPppLuhv5NJd+VSErsDfaMxjJvSmuKuY5KuVQ+le0JknaT5Pyiqs3Ed4e8yHt2V3cmQ+83nBVqd9xDDzDX6s+L+w5XcrO0jkfovh30oRCauwW1c3JraGkER3ovT3bz5BQeZfq8DY3pCX2C09RY7wjJuaoGTypffE895TWdVXi8L76lc1KTvl/ldo91Ufm+MtJ9J4u7Q8i71PcPsD9sjhvoW0Utd/fI6zgkDclnqm9pKbejz9KrkDbRuxxZRa+7cKaf16RObQPiUjeZmxLENB+W4xXHdzmy+sfBx0k6vc/cBnL2rrYeXpozq2/WZff91bfQgJ49+nbQGSPNe4Csc9AhJdWquimrU8j+wp2f83eHqoJzd/Lo4Zy+PP78LbKPW+CIOFIedRuIIa1oWcRV99EAMrE+dUW4u6U31Hruxqnw5m2LMnnlarw93aGk7rpbV9vV5vy/WIuD051687t6483z6ssX+h+ue+Ir8Ug8IT/tipdUmUfkV0v8Lv4W/4h/K27lt8oflT8V6907WqYicq/KX/8D1rXedg==</latexit>

Brady Neal

0.17  E[Y (1)Y (0)]  0Bounds: Monotone Treatment Response

16 / 48

Question:
Given, the nonpositive MTR assumption,
prove E[Y (1) Y (0)]  0.
<latexit sha1_base64="wbg7y8nlyz7QZE28jivuLK57b2E=">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</latexit>

Bounds
No-Assumptions Bound
Monotone Treatment Response
Monotone Treatment Selection
Optimal Treatment Selection

Sensitivity Analysis
Linear Single Confounder
Towards More General Settings

Brady Neal

Bounds: Monotone Treatment Selection

18 / 48

Monotone Treatment Selection (MTS)

Brady Neal

Bounds: Monotone Treatment Selection

19 / 48

Monotone Treatment Selection (MTS)
Treatment groups’ potential outcomes are better than control groups’:E[Y (1) | T = 1]
<latexit sha1_base64="3+0NNp7Va2UwbFKIbEum9tJq0WE=">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</latexit>

Brady NealE[Y (1) | T = 0] ,E[Y (0) | T = 1]

Bounds: Monotone Treatment Selection

E[Y (0) | T = 0]

19 / 48

Monotone Treatment Selection (MTS)
Treatment groups’ potential outcomes are better than control groups’:E[Y (1) | T = 1]
<latexit sha1_base64="3+0NNp7Va2UwbFKIbEum9tJq0WE=">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</latexit>

E[Y (1) | T = 0] ,E[Y (0) | T = 1]

E[Y (0) | T = 0]

Under the MTS assumption, the ATE is bounded from above by
the associational difference. Mathematically,<latexit sha1_base64="lNyM0EyAJy0JRbGf5GrdjbZR8S4=">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</latexit>

Brady NealE[Y (1)

Y (0)]  E[Y | T = 1]

E[Y | T = 0]

Bounds: Monotone Treatment Selection

19 / 48

Monotone Treatment Selection (MTS)
Treatment groups’ potential outcomes are better than control groups’:E[Y (1) | T = 1]
<latexit sha1_base64="3+0NNp7Va2UwbFKIbEum9tJq0WE=">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</latexit>

E[Y (1) | T = 0] ,E[Y (0) | T = 1]

E[Y (0) | T = 0]

Under the MTS assumption, the ATE is bounded from above by
the associational difference. Mathematically,<latexit sha1_base64="lNyM0EyAJy0JRbGf5GrdjbZR8S4=">AAAUI3icrVhLb9tGEN6kr8R9xImOvbBRWiSArEpuCuciIIDtoCgawCn8aiPDkMiVRIivkJRVR9DP6a8peil66aG/otcW6My3S5HUg5QCS5C4mp35ZnZeu6tu4NhR3Gj8dev2e+9/8OFHd+5uffzJp5/d277/4DTyR6EpT0zf8cPzbieSju3Jk9iOHXkehLLjdh151h3u8/zZlQwj2/eO4+tAXridvmf3bLMTE+lyu98+fN32R7Hpu/Jx84mxY8x+NZ5cGF+1HfnGyPAYbde2jHZMOmJXerHRMpoXLFXI0ri43K426g28jMVBUw+qQr+O/Pt3fxRtYQlfmGIkXCGFJ2IaO6IjInq/Fk3REAHRLsSEaCGNbMxLMRVbJDsiLkkcHaIO6btPv15rqke/GTOCtElaHPqEJGmILzWPReMeqOrJ+o0M7yodE2Czjdf07GpMl6ixGBC1TC7hXF+uS2+3ZNUxreEZVmvTSgJQ2A9mbs09ejr0O6YV8vc1cUoaWSQV0sgkmkNURWEdIT2V59k3A0SiAz5JI7a6yO4iq3nep/eQsDo0jmAp22qIFzouHjRL2Mo8DmK6GvEXWqGyrwirN1uDjbirVTDvMVGG4i2N8shFOrPZtZor1sjTJbqYPwaHR9wRUXxkvk3+s4kjn6smYbKODiLZx1oCZFBdI3+P+SSbA4rnDuyJEDUD+WZDT6ArK7U7sdOhZxfYIclPaG4APeyLGtkjgR0iexMv1sAd0q8xfpmw0Zyj11GZHM8arYmzq0a4Ps3YMyzlCeXPxCJl3YRohn7viB+gX1Lca/AK53WNZH1YqLI4gi5Px75FfYX94tI3U9mzCaUGHS7sGCBbhkRL9Skk1vc1rSJC/TgadUJPB14LgFKDdvbPGGMXa+KIjqB7TGMLeljaoF5XF3vajmlOK/vGRnda1Gog/7h2VWec18pyHirV0FnZgqaGeKpXzBFgu7NeM3U3VD05wuqmc1Z1YX/aNThH8/axVBqffKwsZOMAUnlc9uLydS5byy7qpIZYsu/7xNGCVE/TI9RmoDVtZepvX1eJj3iYeLo6J1S9s7bx3MyE6K72lEN8qs55T+KZN4TZQU614b3sfBHioZZPdgPu+BNQi+VOUVl52ZBGk9lMsbyNkYR/1IgjrvAC+DrAWr6YfYzMzLrYN4PnLSByxsYaaV10C7mxfOXHokq9oEpZlfcnR5opIU4nAY0fzTgfES7zDqHFIwpX+GQ2Py2NwQFRptpjPcwldZjaYKPzWeBs68xTHKqi4jkOrupirZyXy/JG0YtlrVm+52UVvViWKVfITVv3gwhY5yVyMeLAEm4mS49LpHzsCYqubP5pDfs87LUjnRehtvCsRNJFj1Gr8nVVviy1L0YXDLVPWObnd/DgFfrfFCegTf2YysYbeTOVu34nn6by4w09m0q6G/o3lXxbIimxN9gzGsu8Kq0p5joq5VL5VLYnSNpNkvOLqrYavrvkRd6zLd2ZDL3fcFao3bGFHlBbqz8v7jtcyc3SOh6h+3bRh0Jotgpq5+TG0NIIjvRenuzmySk8yvR5GxrTE/oEp6mx3hGSc9UOPKl88R31lJd0VuHxvviGzklN+n6R2z3WReX7ykj3nSzuLiHvUd8/wP6wOW6gbxU7ubtHXschaUg+U31LS7kdfZZehbSJ3uXIKnrWwpl+XpM6tQ2IS91krksQ03xYjlcc3+XI6h8HHyfp9D5zE8jZu9p6eGnOrL5Zl933V99CA3r26NtBZ4w07wGyzkGHlFSr6qasTiH7C3d+zt9dqgrO3cnDB3P68vjzt8g+boEj4kh51G0ghrSiZRFX3UcDyMT61BXh7pbeUOu5G6fCm7ctyuSVq/FaukNJ3XW3Lrerzfl/sRYHp7v15rf1xqun1efP9D9cd8Tn4qF4TH7aE8+pMo/Ir6b4Xfwj/hX/VX6t/Fb5o/KnYr19S8tURO5V+ft/Vyj1AQ==</latexit>

E[Y (1)

Y (0)]  E[Y | T = 1]

E[Y | T = 0]

Question: Prove the above MTS upper bound.
Brady Neal

Bounds: Monotone Treatment Selection

19 / 48

Running Example: MTS and MTR
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

Brady NealE[Y | T = 1] = .9

<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

Bounds: Monotone Treatment Selection

20 / 48

Running Example: MTS and MTR
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

No-assumptions bound:

Brady Neal<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit><latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

0.17  E[Y (1)Y (0)]  0.83

Bounds: Monotone Treatment Selection

20 / 48

Running Example: MTS and MTR
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

No-assumptions bound:<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit>

Brady NealE[Y | T = 0] = .2

0.17  E[Y (1)MTS upper bound: E[Y (1)<latexit sha1_base64="lNyM0EyAJy0JRbGf5GrdjbZR8S4=">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</latexit><latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

Y (0)]  0.83

Y (0)]  E[Y | T = 1]

Bounds: Monotone Treatment Selection

E[Y | T = 0]

20 / 48

Running Example: MTS and MTR
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

No-assumptions bound:<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">AAAT7nicrVjbbttGEN0kvSTuzYke88JGKZAAkio5KewXAwFsB0XRAE7hW2sZhkStLEK8haSsOoR+o+hL0dd+RV/bX+jfdObsUiR1IaXAEiSuZmfOzM5td9X1bSuMms3/7ty999HHn3x6/8HGZ59/8eVXmw8fnYTeKDDlsenZXnDW7YTStlx5HFmRLc/8QHacri1Pu8M9nj+9lkFoee5RdOPLC6dz5Vp9y+xERLrc3Ko3G61to23Ld0b74LztjSLTc+Sz1nOjbkx/NZ9fKJZmY+fF5Wa12WjiZcwPWnpQFfp16D188JNoi57whClGwhFSuCKisS06IqT3uWiJpvCJdiFiogU0sjAvxURskOyIuCRxdIg6pO8r+nWuqS79ZswQ0iZpsekTkKQhvtE8PRr3QVVP1m9keJfpiIHNNt7Qs6sxHaJGYkDUMrmEc3W5Lr2dklVHtIYdrNailfigsB/M3Jr79LTpd0Qr5O8b4pQ06pFUQCOTaDZRFYV1BPRUnmffDBCJDvgkjdjqIruLrOZ5j95DwurQOISlbKshXuu4uNAsYSvz2IjpcsRfaYXKviKs/nQNFuKuVsG8R0QZivc0yiMX6cxm13KuSCNPFuhi/ggcLnGHRPGQ+Rb5zyKOfK6ahMk6OojkFdbiI4MaGvkHzCfZ7FM867AnRNQM5JsFPb6urNTuxE6bnl1gByQf09wAetgXNbJHAjtA9iZerIE7oF9j/DJhozlDb6AyOZ41WhNnV41wPZqxpljKE8qfiUXKuphohn7XxY/QLynuNXiF87pGsh4sVFkcQperY79LfYX94tA3U9mzCaUGHQ7sGCBbhkRL9Skk1vctrSJE/dgaNaanDa/5QKlBO/tnjLGDNXFER9A9pnEPeljaoF7XENvajklOK/vGQnea12og/7h2VWec1cpyLirV0Fm5C01N8VKvmCPAdme9ZupuqHpyiNVNZqzqwv60a3CO5u1jqTQ++Vj1kI0DSOVx2YuL17loLVuokxpiyb6/Io5dSPU1PURt+lrTRqb+9nSVeIiHiaejc0LVO2sbz8zERHe0p2ziU3XOexLPvCPMDnKqDe9l54sQD7R8shtwx49BLZY7QWXlZQMaxdOZYnkLIwn/qBFHXOH58LWPtXw9/RiZmVWxbwfPnUPkjI000qroPeTG4pUfiSr1giplVd6fHGmmBDid+DR+OuV8SrjMO4QWlyhc4fF0flIag32iTLTH+phL6jC1wULn64GzrTNPcaiKimY4uKqLtXJeLsobRS+W7U3zPS+r6MWyTLlGblq6H4TAOiuRixAHlnAyWXpUIuVhT1B0ZfPPK9jnYq8d6bwItIWnJZIOeoxalaer8k2pfRG6YKB9wjK/fIAHr9H/JjgBrevHVDZay5up3M0H+TSVH6/p2VTSWdO/qeT7EkmJvcGa0ljmbWlNMddhKZfKp7I9QdJukpxfVLXV8N0lL/Ke3dOdydD7DWeF2h130QNqK/Xn+X2HK7lVWscjdN8u+lAAzb2C2jm+NbQ0giO9lye7eXIKDzN93oLG9IQe4zQ11jtCcq6qw5PKF99TT3lDZxUe74kXdE5q0ffr3O6xKirfV0a672Rxtwh5m/r+PvaH9XF9fauo5+4eeR0HpCH5TPQtLeW29Vl6GdI6ehcjq+j15s70s5rUqW1AXOomc1OCmObDYrzi+C5GVv84eDhJp/eZ20DO3tVWw0tzZvnNuuy+v/wW6tOzT982OmOoefeRdTY6pKRaVTdldQrZm7vzc/5uUVVw7sZPHs3oy+PP3iKvcAscEUfKo24DEaQVLYu47D7qQybSp64Qd7f0htrI3TgV3qxtYSavHI23qzuU1F1343Kz2pr9F2t+cLLVaH3XaL59WX21o//hui8eiyfiGflpW7yiyjwkv5rid/G3+Ef8W/Erv1X+qPypWO/e0TIVkXtV/vofN6/fLQ==</latexit>

E[Y | T = 0] = .2

0.17  E[Y (1)MTS upper bound: E[Y (1)<latexit sha1_base64="lNyM0EyAJy0JRbGf5GrdjbZR8S4=">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</latexit><latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

Y (0)]  0.83

Y (0)]  E[Y | T = 1]

E[Y | T = 0]

Combining MTS upper bound with no-assumptions lower bound:<latexit sha1_base64="0usSocIttC6OGe+olDbQ8S8pVBM=">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</latexit>

Brady Neal

0.17  E[Y (1)Y (0)]  0.7

Bounds: Monotone Treatment Selection

20 / 48

Running Example: MTS and MTR
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

No-assumptions bound:<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit>

E[Y | T = 0] = .2

0.17  E[Y (1)MTS upper bound: E[Y (1)<latexit sha1_base64="lNyM0EyAJy0JRbGf5GrdjbZR8S4=">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</latexit><latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

Y (0)]  0.83

Y (0)]  E[Y | T = 1]

E[Y | T = 0]

Combining MTS upper bound with no-assumptions lower bound:<latexit sha1_base64="0usSocIttC6OGe+olDbQ8S8pVBM=">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</latexit>

0.17  E[Y (1)Y (0)]  0.7

Adding nonnegative MTR assumption and combining MTS upper boundwith MTR lower bound ( E[Y (1) Y (0)] 0 ):<latexit sha1_base64="JsGVuWfln6Und8WBFwah2VbJN/k=">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</latexit>

0  E[Y (1)<latexit sha1_base64="ipIjpYI4RandQpNoFP9mvN7fmeY=">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</latexit>

Brady Neal

Y (0)]  0.7

Bounds: Monotone Treatment Selection

20 / 48

Bounds
No-Assumptions Bound
Monotone Treatment Response
Monotone Treatment Selection
Optimal Treatment Selection

Sensitivity Analysis
Linear Single Confounder
Towards More General Settings

Brady Neal

Bounds: Optimal Treatment Selection

21 / 48

Optimal Treatment Selection (OTS) Assumption

Brady Neal

Bounds: Optimal Treatment Selection

22 / 48

Optimal Treatment Selection (OTS) Assumption
Individuals always receive the treatment that is best for them:

Brady Neal

Bounds: Optimal Treatment Selection

22 / 48

Optimal Treatment Selection (OTS) Assumption
Individuals always receive the treatment that is best for them:<latexit sha1_base64="y9WY9C9eeEOVDI3ZnSJL/kkEjgs=">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</latexit>

Ti = 1 =) Yi (1)Brady Neal

Yi (0) ,Ti = 0 =) Yi (0) > Yi (1)Bounds: Optimal Treatment Selection

22 / 48

Optimal Treatment Selection (OTS) Assumption
Individuals always receive the treatment that is best for them:<latexit sha1_base64="y9WY9C9eeEOVDI3ZnSJL/kkEjgs=">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</latexit>

Ti = 1 =) Yi (1)<latexit sha1_base64="uOahBpzZ6+fBLKjwT1Oth/i4FKs=">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</latexit>

Brady Neal

Yi (0) ,Ti = 0 =) Yi (0) > Yi (1)E[Y (1) | T = 0]  E[Y (0) | T = 0] = E[Y | T = 0]

Bounds: Optimal Treatment Selection

22 / 48

Optimal Treatment Selection (OTS) Assumption
Individuals always receive the treatment that is best for them:<latexit sha1_base64="y9WY9C9eeEOVDI3ZnSJL/kkEjgs=">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</latexit>

Ti = 1 =) Yi (1)<latexit sha1_base64="Eui1M4nGH2vBrYURJfeCmLq9mbc=">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</latexit>

Brady Neal

Yi (0) ,Ti = 0 =) Yi (0) > Yi (1)E[Y (1) | T = 0]  E[Y | T = 0]

Bounds: Optimal Treatment Selection

22 / 48

Optimal Treatment Selection (OTS) Assumption
Individuals always receive the treatment that is best for them:<latexit sha1_base64="y9WY9C9eeEOVDI3ZnSJL/kkEjgs=">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</latexit>

Ti = 1 =) Yi (1)<latexit sha1_base64="Eui1M4nGH2vBrYURJfeCmLq9mbc=">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</latexit><latexit sha1_base64="ltRyO403K+oXpHo6pnpnFTqEeqg=">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</latexit>

Brady Neal

Yi (0) ,Ti = 0 =) Yi (0) > Yi (1)E[Y (1) | T = 0]  E[Y | T = 0]
E[Y (0) | T = 1]  E[Y (1) | T = 1] = E[Y | T = 1]

Bounds: Optimal Treatment Selection

22 / 48

Optimal Treatment Selection (OTS) Assumption
Individuals always receive the treatment that is best for them:<latexit sha1_base64="y9WY9C9eeEOVDI3ZnSJL/kkEjgs=">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</latexit>

Ti = 1 =) Yi (1)<latexit sha1_base64="Eui1M4nGH2vBrYURJfeCmLq9mbc=">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</latexit><latexit sha1_base64="7U50vAPgukfnG7eclSmxFT/yXpc=">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</latexit>

Brady Neal

Yi (0) ,Ti = 0 =) Yi (0) > Yi (1)E[Y (1) | T = 0]  E[Y | T = 0]
E[Y (0) | T = 1]  E[Y | T = 1]

Bounds: Optimal Treatment Selection

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OTS Upper Bound 1

Brady Neal

Bounds: Optimal Treatment Selection

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OTS Upper Bound 1
OTS assumption tells us that E[Y (1) | T = 0]  E[Y | T = 0]<latexit sha1_base64="Eui1M4nGH2vBrYURJfeCmLq9mbc=">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</latexit>

Brady Neal

Bounds: Optimal Treatment Selection

23 / 48

OTS Upper Bound 1
OTS assumption tells us that E[Y (1) | T = 0]  E[Y | T = 0]<latexit sha1_base64="Eui1M4nGH2vBrYURJfeCmLq9mbc=">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</latexit><latexit sha1_base64="rduDf5k8iF6yBenHg9Dav7eKUUw=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]

 ⇡ E[Y | T = 1] + (1
⇡a

(1

(1

⇡) E[Y | T = 0]

(Observational-Counterfactual
Decomposition)⇡) E[Y | T = 0]⇡) E[Y | T = 0]

= ⇡ E[Y | T = 1]

Brady Neal⇡) E[Y (1) | T = 0]⇡aBounds: Optimal Treatment Selection

23 / 48

OTS Upper Bound 1
OTS assumption tells us that E[Y (1) | T = 0]  E[Y | T = 0]<latexit sha1_base64="Eui1M4nGH2vBrYURJfeCmLq9mbc=">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</latexit><latexit sha1_base64="rduDf5k8iF6yBenHg9Dav7eKUUw=">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</latexit><latexit sha1_base64="rduDf5k8iF6yBenHg9Dav7eKUUw=">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</latexit>

E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

 ⇡ E[Y | T = 1] + (1 ⇡) E[Y | T = 0]
⇡ a (1 ⇡) E[Y | T = 0]
= ⇡ E[Y | T = 1]

Brady Neal⇡aBounds: Optimal Treatment Selection

23 / 48

OTS Upper Bound 1
OTS assumption tells us that E[Y (1) | T = 0]  E[Y | T = 0]<latexit 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E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

 ⇡ E[Y | T = 1] + (1 ⇡) E[Y | T = 0]
⇡ a (1 ⇡) E[Y | T = 0]
= ⇡ E[Y | T = 1]

Brady Neal⇡aBounds: Optimal Treatment Selection

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OTS Lower Bound 1

Brady Neal

Bounds: Optimal Treatment Selection

24 / 48

OTS Lower Bound 1
OTS assumption tells us that E[Y (0) | T = 1]  E[Y | T = 1]
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Brady Neal

Bounds: Optimal Treatment Selection

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OTS Lower Bound 1
OTS assumption tells us that E[Y (0) | T = 1]
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Brady Neal

Bounds: Optimal Treatment Selection

E[Y | T = 1]

24 / 48

OTS Lower Bound 1
OTS assumption tells us that E[Y (0) | T = 1]
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E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]

⇡ E[Y | T = 1] + (1
⇡ E[Y | T = 1]

= (1Brady Neal

⇡) a

(1

(1

E[Y | T = 1]⇡) E[Y (1) | T = 0]
(1

⇡) a

⇡) E[Y | T = 0]

(Observational-Counterfactual
Decomposition)⇡) E[Y | T = 0]⇡) E[Y | T = 0]

Bounds: Optimal Treatment Selection

24 / 48

OTS Lower Bound 1
OTS assumption tells us that E[Y (0) | T = 1]
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E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]

E[Y | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

⇡ E[Y | T = 1] + (1 ⇡) a
⇡ E[Y | T = 1] (1 ⇡) E[Y | T = 0]

= (1Brady Neal

⇡) a

(1

⇡) E[Y | T = 0]

Bounds: Optimal Treatment Selection

24 / 48

OTS Lower Bound 1
OTS assumption tells us that E[Y (0) | T = 1]
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E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]

E[Y | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

⇡ E[Y | T = 1] + (1 ⇡) a
⇡ E[Y | T = 1] (1 ⇡) E[Y | T = 0]

= (1Brady Neal

⇡) a

(1

⇡) E[Y | T = 0]

Bounds: Optimal Treatment Selection

24 / 48

OTS Complete Bound 1 and Running Example

Brady Neal

Bounds: Optimal Treatment Selection

25 / 48

OTS Complete Bound 1 and Running Example<latexit sha1_base64="FdLyHQl37K3hksq2KLHFOt36Cbg=">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</latexit>

E[Y (1)
E[Y (1)

Y (0)] < ⇡ E[Y | T = 1]
Y (0)]

(1

⇡) a

(1

⇡a
⇡) E[Y | T = 0]

Interval Length = ⇡ E[Y | T = 1] + (1Brady Neal⇡) E[Y | T = 0]

Bounds: Optimal Treatment Selectiona25 / 48

OTS Complete Bound 1 and Running Example<latexit sha1_base64="FdLyHQl37K3hksq2KLHFOt36Cbg=">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</latexit>

E[Y (1)

Y (0)] < ⇡ E[Y | T = 1]

E[Y (1)

Y (0)]

(1

⇡) a

(1

⇡a
⇡) E[Y | T = 0]

Interval Length = ⇡ E[Y | T = 1] + (1⇡) E[Y | T = 0]aRunning example
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

Brady NealE[Y | T = 1] = .9

<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

Bounds: Optimal Treatment Selection

25 / 48

OTS Complete Bound 1 and Running Example<latexit sha1_base64="FdLyHQl37K3hksq2KLHFOt36Cbg=">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</latexit>

E[Y (1)

Y (0)] < ⇡ E[Y | T = 1]

E[Y (1)

Y (0)]

(1

⇡) a

(1

⇡a
⇡) E[Y | T = 0]

Interval Length = ⇡ E[Y | T = 1] + (1⇡) E[Y | T = 0]aRunning example
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

No-assumptions bound:

Brady Neal<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">AAAT7nicrVjbbttGEN0kvSTuzYke88JGKZAAkio5KewXAwFsB0XRAE7hW2sZhkStLEK8haSsOoR+o+hL0dd+RV/bX+jfdObsUiR1IaXAEiSuZmfOzM5td9X1bSuMms3/7ty999HHn3x6/8HGZ59/8eVXmw8fnYTeKDDlsenZXnDW7YTStlx5HFmRLc/8QHacri1Pu8M9nj+9lkFoee5RdOPLC6dz5Vp9y+xERLrc3Ko3G61to23Ld0b74LztjSLTc+Sz1nOjbkx/NZ9fKJZmY+fF5Wa12WjiZcwPWnpQFfp16D188JNoi57whClGwhFSuCKisS06IqT3uWiJpvCJdiFiogU0sjAvxURskOyIuCRxdIg6pO8r+nWuqS79ZswQ0iZpsekTkKQhvtE8PRr3QVVP1m9keJfpiIHNNt7Qs6sxHaJGYkDUMrmEc3W5Lr2dklVHtIYdrNailfigsB/M3Jr79LTpd0Qr5O8b4pQ06pFUQCOTaDZRFYV1BPRUnmffDBCJDvgkjdjqIruLrOZ5j95DwurQOISlbKshXuu4uNAsYSvz2IjpcsRfaYXKviKs/nQNFuKuVsG8R0QZivc0yiMX6cxm13KuSCNPFuhi/ggcLnGHRPGQ+Rb5zyKOfK6ahMk6OojkFdbiI4MaGvkHzCfZ7FM867AnRNQM5JsFPb6urNTuxE6bnl1gByQf09wAetgXNbJHAjtA9iZerIE7oF9j/DJhozlDb6AyOZ41WhNnV41wPZqxpljKE8qfiUXKuphohn7XxY/QLynuNXiF87pGsh4sVFkcQperY79LfYX94tA3U9mzCaUGHQ7sGCBbhkRL9Skk1vctrSJE/dgaNaanDa/5QKlBO/tnjLGDNXFER9A9pnEPeljaoF7XENvajklOK/vGQnea12og/7h2VWec1cpyLirV0Fm5C01N8VKvmCPAdme9ZupuqHpyiNVNZqzqwv60a3CO5u1jqTQ++Vj1kI0DSOVx2YuL17loLVuokxpiyb6/Io5dSPU1PURt+lrTRqb+9nSVeIiHiaejc0LVO2sbz8zERHe0p2ziU3XOexLPvCPMDnKqDe9l54sQD7R8shtwx49BLZY7QWXlZQMaxdOZYnkLIwn/qBFHXOH58LWPtXw9/RiZmVWxbwfPnUPkjI000qroPeTG4pUfiSr1giplVd6fHGmmBDid+DR+OuV8SrjMO4QWlyhc4fF0flIag32iTLTH+phL6jC1wULn64GzrTNPcaiKimY4uKqLtXJeLsobRS+W7U3zPS+r6MWyTLlGblq6H4TAOiuRixAHlnAyWXpUIuVhT1B0ZfPPK9jnYq8d6bwItIWnJZIOeoxalaer8k2pfRG6YKB9wjK/fIAHr9H/JjgBrevHVDZay5up3M0H+TSVH6/p2VTSWdO/qeT7EkmJvcGa0ljmbWlNMddhKZfKp7I9QdJukpxfVLXV8N0lL/Ke3dOdydD7DWeF2h130QNqK/Xn+X2HK7lVWscjdN8u+lAAzb2C2jm+NbQ0giO9lye7eXIKDzN93oLG9IQe4zQ11jtCcq6qw5PKF99TT3lDZxUe74kXdE5q0ffr3O6xKirfV0a672Rxtwh5m/r+PvaH9XF9fauo5+4eeR0HpCH5TPQtLeW29Vl6GdI6ehcjq+j15s70s5rUqW1AXOomc1OCmObDYrzi+C5GVv84eDhJp/eZ20DO3tVWw0tzZvnNuuy+v/wW6tOzT982OmOoefeRdTY6pKRaVTdldQrZm7vzc/5uUVVw7sZPHs3oy+PP3iKvcAscEUfKo24DEaQVLYu47D7qQybSp64Qd7f0htrI3TgV3qxtYSavHI23qzuU1F1343Kz2pr9F2t+cLLVaH3XaL59WX21o//hui8eiyfiGflpW7yiyjwkv5rid/G3+Ef8W/Erv1X+qPypWO/e0TIVkXtV/vofN6/fLQ==</latexit><latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

0.17  E[Y (1)Y (0)]  0.83

Bounds: Optimal Treatment Selection

25 / 48

OTS Complete Bound 1 and Running Example<latexit sha1_base64="FdLyHQl37K3hksq2KLHFOt36Cbg=">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</latexit>

E[Y (1)

Y (0)] < ⇡ E[Y | T = 1]

E[Y (1)

Y (0)]

(1

⇡) a

(1

⇡a
⇡) E[Y | T = 0]

Interval Length = ⇡ E[Y | T = 1] + (1⇡) E[Y | T = 0]aRunning example
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

No-assumptions bound:
OTS Bound 1:<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit><latexit sha1_base64="XMUf2rSbtZk24I7/igjQCRTQueM=">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</latexit><latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

0.17  E[Y (1)Y (0)]  0.83

0.14  E[Y (1)Y (0)]  0.27

Interval Length = 0.41

Brady Neal

Bounds: Optimal Treatment Selection

25 / 48

Bound that identifies the sign

OTS Upper Bound 2 Preliminaries

Brady Neal

Bounds: Optimal Treatment Selection

27 / 48

OTS Upper Bound 2 Preliminaries
OTS Assumption: Ti = 1 =) Yi (1)<latexit sha1_base64="y9WY9C9eeEOVDI3ZnSJL/kkEjgs=">AAAUQXicrVhLb9tGEF6nr8R92fGtubBxCqQArUpuCufiIoCdoCgawCmcxG0UGBJF2YT5MknZdQQd+8v6K/oTil6KXnvpN98uRVIvykEkSFzNznwzO6/dVTf2vTRrNv9cufHe+x98+NHNW6sff/LpZ5+vrd9+kUaDxHGfO5EfJUfdTur6Xug+z7zMd4/ixO0EXd992T3bk/mXF26SelF4mF3F7uugcxJ6fc/pZCAdr/3ezsCeBW6YHXvWrtWy2l4AxW5qtaNB5kSBe+zdb31ttU/c8zKpCZJt2asWXu3zQaenR1W05kw0iH5fRT9e22w2mnxZ04OWGWwq8zqI1m/9rNqqpyLlqIEKlKtClWHsq45K8X6lWqqpYtBeqyFoCUYe5101UquQHYDLBUcH1DN8n+DXK0MN8VswU0o70OLjk0DSUl8Znh7GfVL1U/RbJd55OobEFhuv8OwazADUTJ2CWieXcy4v18U7qFl1hjU85Go9rCQmRfzgVNbcx9PH7wwrlO8rcLoY9SCVYOSA5oOqKaIjwVN7Xnxzykh0yOdiJFYvsnuR1TIf4X0GrA7GKS0VWy31xMQlpGaXtgqPz5jOR/wNK9T2LcLqj9fgMe56FcJ7CMqZeoNRFXmRznJ2zefKDPJohi7hz8gRgjsFJWLme/CfB45qrjrAFB0dRvKEa4mZQQ2D/CPn82yOEc8t2pMyahbzzaOe2FRWYXdup49nl9gJ5IeYO6Ue8YUNe1xiJ8ze3Is2uRP8uuQvhzY6E/QGK1PiaWNNkl02cCPMeGMs7Qntz9wibd0QNMu8t9RP1O8i7ja9InltQzaihTqLU+oKTex30VfELwG+hSqezSk2dQS045TZcgZaoU8jib5vsIqU9eMb1CGePr0WE8WmdvHPJccB1yQRHVD3JcY96hFpC72uoXaMHaOKVvGNx+40rdVi/knt6s44qVXkQlaqZbJyl5qa6oFZsURA7C57zTHdUPfklKsbTVjVpf1F15AcrdonUkV8qrHqMRtPKVXFFS/OXuestWyzTmzGUnx/Ao5dSvUNPWVtxkbTaqn+9kyVRIyHw2dgckLXu2i7nJgZgh4YT/ng03Uue5LMnAOzw5xq03vl+UWIj418vhtIxx+SuljuBSurKptgNBzPLJb3OHLpHz2SiGu8mL6OuZYvxx+rNLMs9rvBC6cQJWMzg7Qseo+5MXvlh2oTvWATWVX1p0RaKAlPJzHG98ac94ArvGfUEoIiFT4cz49qY7APysh4rM+5vA4LGzx2vh452ybzNIeuqGyCQ6p6sVbJy1l5o+mLZXvjfK/KavpiWaFcMDc90w9SYh3VyGWMg0gEpSw9rJGKuCdourb5lyXsC7nXDkxeJMbClzWSAXuMXlVkqvJprX0Zu2BifCIyv76FBy/Y/0Y8AV3Xj4Vsdi1vFnJXb+XTQv7ymp4tJINr+reQfFMj6XJv8MY0kXlWW1PCdVDLpfOpbk9wsZvk5xddbTa/u/Ci7Nk905kss99IVujdcZc9wF6qP0/vO1LJrdo6HrD7dtmHEmruLaid5+8MrYjgwOzl+W6en8LTUp/3qLE4oQ95mro0O0J+rtqiJ7UvfkBPeYqzioz31Lc4J7Xw/aSyeyyLKveVgek7ZdxtIO+g7+9zf7g+bmxuFVuVu0dVx2NoyD8jc0sruH1zlp6HdB29s5F19HpTZ/pJTfrUdgoufZO5qkEs8mE23uL4zkbW/zhEPEkX95l3gVy+qy2HV+TM/Jt13X1//i00xrOPb5+dMTW8+8w6nx3SRa3qm7I+hexN3fklf7dRFZK7w7u3J/RV8SdvkSe8BQ7AUfDo20BGaU0rI867j8aUycypK+XdrbihNio3To03aVtayqvA4O2aDuWarrt6vLbZmvwXa3rwYrvR+q7RfPZg89FD8w/XTXVH3VX34acd9QiVeQC/OurvlfWVL1bubPyx8dfGPxv/atYbK0ZmQ1VeG//9D9ub+9I=</latexit>

Brady Neal

Yi (0) ,Ti = 0 =) Yi (0) > Yi (1)Bounds: Optimal Treatment Selection

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OTS Upper Bound 2 Preliminaries
OTS Assumption: Ti = 1 =) Yi (1)<latexit sha1_base64="y9WY9C9eeEOVDI3ZnSJL/kkEjgs=">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</latexit>

Yi (0) ,Ti = 0 =) Yi (0) > Yi (1)OTS Bound 1 implication we used: E[Y (1) | T = 0]  E[Y | T = 0]<latexit sha1_base64="Eui1M4nGH2vBrYURJfeCmLq9mbc=">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</latexit>

Brady Neal

Bounds: Optimal Treatment Selection

27 / 48

OTS Upper Bound 2 Preliminaries
OTS Assumption: Ti = 1 =) Yi (1)<latexit sha1_base64="y9WY9C9eeEOVDI3ZnSJL/kkEjgs=">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</latexit>

Yi (0) ,Ti = 0 =) Yi (0) > Yi (1)OTS Bound 1 implication we used: E[Y (1) | T = 0]  E[Y | T = 0]<latexit sha1_base64="Eui1M4nGH2vBrYURJfeCmLq9mbc=">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</latexit>

OTS Bound 2 implication we’ll use: E[Y (1) | T = 0]  E[Y | T = 1]
<latexit sha1_base64="M7DwQ7lnJFzJ8eNGP6ZfH6MVtHo=">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</latexit>

Brady Neal

Bounds: Optimal Treatment Selection

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OTS Upper Bound 2 Preliminaries
OTS Assumption: Ti = 1 =) Yi (1)<latexit sha1_base64="y9WY9C9eeEOVDI3ZnSJL/kkEjgs=">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</latexit>

Yi (0) ,Ti = 0 =) Yi (0) > Yi (1)OTS Bound 1 implication we used: E[Y (1) | T = 0]  E[Y | T = 0]<latexit sha1_base64="Eui1M4nGH2vBrYURJfeCmLq9mbc=">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</latexit>

OTS Bound 2 implication we’ll use: E[Y (1) | T = 0]  E[Y | T = 1]
<latexit sha1_base64="M7DwQ7lnJFzJ8eNGP6ZfH6MVtHo=">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</latexit>

Proof: E[Y (1) | T = 0] = E[Y (1) | Y (0) > Y (1)]<latexit sha1_base64="EjjEZ8AnvzKw6QZ52nI+ep+IfOA=">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</latexit>

 E[Y (1) | Y (0)  Y (1)]= E[Y (1) | T = 1]
= E[Y | T = 1]

Brady Neal

Bounds: Optimal Treatment Selection

27 / 48

OTS Upper Bound 2 Preliminaries
OTS Assumption: Ti = 1 =) Yi (1)<latexit sha1_base64="y9WY9C9eeEOVDI3ZnSJL/kkEjgs=">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</latexit>

Yi (0) ,Ti = 0 =) Yi (0) > Yi (1)OTS Bound 1 implication we used: E[Y (1) | T = 0]  E[Y | T = 0]<latexit sha1_base64="Eui1M4nGH2vBrYURJfeCmLq9mbc=">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</latexit>

OTS Bound 2 implication we’ll use: E[Y (1) | T = 0]  E[Y | T = 1]
<latexit sha1_base64="M7DwQ7lnJFzJ8eNGP6ZfH6MVtHo=">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</latexit>

Proof: E[Y (1) | T = 0] = E[Y (1) | Y (0) > Y (1)]<latexit sha1_base64="EjjEZ8AnvzKw6QZ52nI+ep+IfOA=">AAAUqXicrVjbbttGEF2lt8S9Oanf+sJGaZCgsiq5KZwXFQFsB0XRAHZrO04sw5DIlUWIt5CUVUfQS/uV/Yj+Q2fOLkVSF1IKLEHianbmzOzcdlfdwLGjuNH4t3Lno48/+fSzu/c2Pv/iy6++3rz/4DTyh6EpT0zf8cOzbieSju3Jk9iOHXkWhLLjdh35ujvY4/nX1zKMbN87jm8CeeF2rjy7Z5udmEiXm/+1D87b/jA2fVc+aT412q5tGe2YIGJXerHRMhoXxuOWsZAtITSeGr8YmekLo93eMOj1uO3Id6WyimmR+BK9WfOaS7gXsl5uVhv1Bl7G/KCpB1WhX4f+/Xt/iLawhC9MMRSukMITMY0d0RERvc9FUzREQLQLMSZaSCMb81JMxAbJDolLEkeHqAP6vqJf55rq0W/GjCBtkhaHPiFJGuJ7zWPRuAeqerJ+I8O7TMcY2GzjDT27GtMlaiz6RC2TSzhXl+vS2y1ZdUxreI7V2rSSABT2g5lbc4+eDv2OaYX8fUOckkYWSYU0MonmEFVRWEdIT+V59k0fkeiAT9KIrS6yu8hqnvfpPSCsDo0jWMq2GuKljosHzRK2Mo+DmC5H/ItWqOwrwupN12Aj7moVzHtMlIF4T6M8cpHObHYt54o18mSBLuaPweERd0QUH5lvk/9s4sjnqkmYrKODSF5hLQEyqK6Rf8N8ks0BxXMb9kSImoF8s6En0JWV2p3Y6dCzC+yQ5Mc014ce9kWN7JHADpG9iRdr4A7p1wi/TNhoztDrqEyOZ43WxNlVI1yfZuwplvKE8mdikbJuTDRDv7fF79AvKe41eIXzukayPixUWRxBl6dj36K+wn5x6Zup7NmEUoMOF3b0kS0DoqX6FBLr+5FWEaF+HI06pqcDrwVAqUE7+2eEsYs1cUSH0D2isQU9LG1Qr6uLXW3HJKeVfWOjO81rNZB/XLuqM85qZTkPlWrorGxBU0M80yvmCLDdWa+ZuhuqnhxhdZMZq7qwP+0anKN5+1gqjU8+VhaysQ+pPC57cfE6F61lB3VSQyzZ91fE0YJUT9Mj1GagNW1k6m9PV4mPeJh4ujonVL2zttHMzJjorvaUQ3yqznlP4pl3hNlBTrXhvex8EeKBlk92A+74Y1CL5U5RWXnZkEbj6UyxvI2RhH/UiCOu8AL4OsBavpt+jMzMqti3g+fNIXLGxhppVXQLubF45ceiSr2gSlmV9ydHmikhTicBjR9NOR8RLvMOoMUjClf4eDo/KY3BPlEm2mM9zCV1mNpgo/NZ4GzrzFMcqqLiGQ6u6mKtnJeL8kbRi2Wtab7nZRW9WJYp18hNW/eDCFhnJXIx4sASbiZLj0ukfOwJiq5sfrOCfR722qHOi1Bb+LpE0kWPUavydVW+KrUvRhcMtU9Y5u0HePAa/W+CE9C6fkxl47W8mcrdfJBPU/nRmp5NJd01/ZtKvi+RlNgb7CmNZY5Ka4q5Dku5VD6V7QmSdpPk/KKqrYbvLnmR92xLdyZD7zecFWp3bKEH1Fbqz/P7Dldys7SOh+i+XfShEJqtgto5uTW0NIJDvZcnu3lyCo8yfd6GxvSEPsZpaqR3hORctQ1PKl/8Sj3lFZ1VeLwnfqJzUpO+X+Z2j1VR+b4y1H0ni7tDyLvU9/exP6yPG+hbxXbu7pHXcUAaks9E39JSbkefpZchraN3MbKKnjV3pp/VpE5tfeJSN5mbEsQ0HxbjFcd3MbL6x8HHSTq9z9wGcvauthpemjPLb9Zl9/3lt9CAnj36dtAZI827j6xz0CEl1aq6KatTyN7cnZ/zd4eqgnN3/PDBjL48/uwt8gq3wCFxpDzqNhBDWtGyiMvuowFkYn3qinB3S2+o9dyNU+HN2hZl8srVeC3doaTuuhuXm9Xm7L9Y84PTnXrz53rj6Fn1xXP9D9dd8a14KJ6Qn3bFC6rMQ/KrWfmzclP5u/LP1g9bR1tnW28V652KlvlG5F5b5v+V7Rlb</latexit>

 E[Y (1) | Y (0)  Y (1)]= E[Y (1) | T = 1]
= E[Y | T = 1]

Brady Neal

Bounds: Optimal Treatment Selection

27 / 48

OTS Upper Bound 2 Preliminaries
OTS Assumption: Ti = 1 =) Yi (1)<latexit sha1_base64="y9WY9C9eeEOVDI3ZnSJL/kkEjgs=">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</latexit>

Yi (0) ,Ti = 0 =) Yi (0) > Yi (1)OTS Bound 1 implication we used: E[Y (1) | T = 0]  E[Y | T = 0]<latexit sha1_base64="Eui1M4nGH2vBrYURJfeCmLq9mbc=">AAAUB3icrVjbbttGEN2kt8S9OdFjX9goBVJAViU3hfMiIIDtoGgbwCl8ay3DkMiVRIi3kJRVR9AH9GuKvhR97VcUfWv/pDNnlyKpCykFliBxNTtzZnZuu6tu4NhR3Gj8c+fuO+++9/4H9+5vffjRx598uv3g4Wnkj0JTnpi+44fn3U4kHduTJ7EdO/I8CGXH7TryrDvc5/mzaxlGtu8dxzeBvHQ7fc/u2WYnJtLV9vdG+/Ci7Y9i03flk+aXRtu1LaMdE0bsSi82Wkbj0mg78rWR5VzKdrVdbdQbeBmLg6YeVIV+HfkP7v8o2sISvjDFSLhCCk/ENHZER0T0vhBN0RAB0S7FhGghjWzMSzEVWyQ7Ii5JHB2iDum7T78uNNWj34wZQdokLQ59QpI0xBeax6JxD1T1ZP1GhneVjgmw2cYbenY1pkvUWAyIWiaXcK4v16W3W7LqmNbwDKu1aSUBKOwHM7fmHj0d+h3TCvn7hjgljSySCmlkEs0hqqKwjpCeyvPsmwEi0QGfpBFbXWR3kdU879N7SFgdGkewlG01xAsdFw+aJWxlHgcxXY34C61Q2VeE1ZutwUbc1SqY95goQ/GGRnnkIp3Z7FrNFWvk6RJdzB+DwyPuiCg+Mt8m/9nEkc9VkzBZRweR7GMtATKorpG/w3ySzQHFcwf2RIiagXyzoSfQlZXandjp0LML7JDkJzQ3gB72RY3skcAOkb2JF2vgDunXGL9M2GjO0euoTI5njdbE2VUjXJ9m7BmW8oTyZ2KRsm5CNEO/d8QP0C8p7jV4hfO6RrI+LFRZHEGXp2Pfor7CfnHpm6ns2YRSgw4XdgyQLUOipfoUEuv7ilYRoX4cjTqhpwOvBUCpQTv7Z4yxizVxREfQPaaxBT0sbVCvq4s9bcc0p5V9Y6M7LWo1kH9cu6ozzmtlOQ+VauisbEFTQzzVK+YIsN1Zr5m6G6qeHGF10zmrurA/7Rqco3n7WCqNTz5WFrJxAKk8Lntx+TqXrWUXdVJDLNn3feJoQaqn6RFqM9CatjL1t6+rxEc8TDxdnROq3lnbeG5mQnRXe8ohPlXnvCfxzGvC7CCn2vBedr4I8VDLJ7sBd/wJqMVyp6isvGxIo8lspljexkjCP2rEEVd4AXwdYC2fzz5GZmZd7NvB8xYQOWNjjbQuuoXcWL7yY1GlXlClrMr7kyPNlBCnk4DGj2ecjwmXeYfQ4hGFK3wym5+WxuCAKFPtsR7mkjpMbbDR+SxwtnXmKQ5VUfEcB1d1sVbOy2V5o+jFstYs3/Oyil4sy5Rr5Kat+0EErPMSuRhxYAk3k6XHJVI+9gRFVzb/tIZ9Hvbakc6LUFt4ViLposeoVfm6Kl+W2hejC4baJyzz81t48Br9b4oT0KZ+TGXjjbyZyt28lU9T+fGGnk0l3Q39m0q+KZGU2BvsGY1lXpXWFHMdlXKpfCrbEyTtJsn5RVVbDd9d8iLv2ZbuTIbebzgr1O7YQg+ordWfF/cdruRmaR2P0H276EMhNFsFtXNya2hpBEd6L0928+QUHmX6vA2N6Ql9gtPUWO8IyblqB55UvviWespLOqvweF98TeekJn2/yO0e66LyfWWk+04Wd5eQ96jvH2B/2Bw30LeKndzdI6/jkDQkn6m+paXcjj5Lr0LaRO9yZBU9a+FMP69JndoGxKVuMjcliGk+LMcrju9yZPWPg4+TdHqfuQ3k7F1tPbw0Z1bfrMvu+6tvoQE9e/TtoDNGmvcAWeegQ0qqVXVTVqeQ/YU7P+fvLlUF5+7k0cM5fXn8+VtkH7fAEXGkPOo2EENa0bKIq+6jAWRifeqKcHdLb6j13I1T4c3bFmXyytV4Ld2hpO66W1fb1eb8v1iLg9PdevObeuPV0+rzZ/ofrnviM/FIPCE/7YnnVJlH5FdT/C7+Fv+K/yq/Vn6r/FH5U7HevaNlKiL3qvz1P0c26p4=</latexit>

OTS Bound 2 implication we’ll use: E[Y (1) | T = 0]  E[Y | T = 1]
<latexit sha1_base64="M7DwQ7lnJFzJ8eNGP6ZfH6MVtHo=">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</latexit>

Proof: E[Y (1) | T = 0] = E[Y (1) | Y (0) > Y (1)]<latexit sha1_base64="EjjEZ8AnvzKw6QZ52nI+ep+IfOA=">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</latexit><latexit sha1_base64="EjjEZ8AnvzKw6QZ52nI+ep+IfOA=">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</latexit>

 E[Y (1) | Y (0)  Y (1)]= E[Y (1) | T = 1]
= E[Y | T = 1]

Brady Neal

Bounds: Optimal Treatment Selection

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OTS Upper Bound 2 Preliminaries
OTS Assumption: Ti = 1 =) Yi (1)<latexit sha1_base64="y9WY9C9eeEOVDI3ZnSJL/kkEjgs=">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</latexit>

Yi (0) ,Ti = 0 =) Yi (0) > Yi (1)OTS Bound 1 implication we used: E[Y (1) | T = 0]  E[Y | T = 0]<latexit sha1_base64="Eui1M4nGH2vBrYURJfeCmLq9mbc=">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</latexit>

OTS Bound 2 implication we’ll use: E[Y (1) | T = 0]  E[Y | T = 1]
<latexit sha1_base64="M7DwQ7lnJFzJ8eNGP6ZfH6MVtHo=">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</latexit>

Proof: E[Y (1) | T = 0] = E[Y (1) | Y (0) > Y (1)]<latexit sha1_base64="EjjEZ8AnvzKw6QZ52nI+ep+IfOA=">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</latexit><latexit sha1_base64="EjjEZ8AnvzKw6QZ52nI+ep+IfOA=">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</latexit><latexit sha1_base64="EjjEZ8AnvzKw6QZ52nI+ep+IfOA=">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</latexit>

 E[Y (1) | Y (0)  Y (1)]= E[Y (1) | T = 1]
= E[Y | T = 1]

Brady Neal

Bounds: Optimal Treatment Selection

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OTS Upper Bound 2 Preliminaries
OTS Assumption: Ti = 1 =) Yi (1)<latexit sha1_base64="y9WY9C9eeEOVDI3ZnSJL/kkEjgs=">AAAUQXicrVhLb9tGEF6nr8R92fGtubBxCqQArUpuCufiIoCdoCgawCmcxG0UGBJF2YT5MknZdQQd+8v6K/oTil6KXnvpN98uRVIvykEkSFzNznwzO6/dVTf2vTRrNv9cufHe+x98+NHNW6sff/LpZ5+vrd9+kUaDxHGfO5EfJUfdTur6Xug+z7zMd4/ixO0EXd992T3bk/mXF26SelF4mF3F7uugcxJ6fc/pZCAdr/3ezsCeBW6YHXvWrtWy2l4AxW5qtaNB5kSBe+zdb31ttU/c8zKpCZJt2asWXu3zQaenR1W05kw0iH5fRT9e22w2mnxZ04OWGWwq8zqI1m/9rNqqpyLlqIEKlKtClWHsq45K8X6lWqqpYtBeqyFoCUYe5101UquQHYDLBUcH1DN8n+DXK0MN8VswU0o70OLjk0DSUl8Znh7GfVL1U/RbJd55OobEFhuv8OwazADUTJ2CWieXcy4v18U7qFl1hjU85Go9rCQmRfzgVNbcx9PH7wwrlO8rcLoY9SCVYOSA5oOqKaIjwVN7Xnxzykh0yOdiJFYvsnuR1TIf4X0GrA7GKS0VWy31xMQlpGaXtgqPz5jOR/wNK9T2LcLqj9fgMe56FcJ7CMqZeoNRFXmRznJ2zefKDPJohi7hz8gRgjsFJWLme/CfB45qrjrAFB0dRvKEa4mZQQ2D/CPn82yOEc8t2pMyahbzzaOe2FRWYXdup49nl9gJ5IeYO6Ue8YUNe1xiJ8ze3Is2uRP8uuQvhzY6E/QGK1PiaWNNkl02cCPMeGMs7Qntz9wibd0QNMu8t9RP1O8i7ja9InltQzaihTqLU+oKTex30VfELwG+hSqezSk2dQS045TZcgZaoU8jib5vsIqU9eMb1CGePr0WE8WmdvHPJccB1yQRHVD3JcY96hFpC72uoXaMHaOKVvGNx+40rdVi/knt6s44qVXkQlaqZbJyl5qa6oFZsURA7C57zTHdUPfklKsbTVjVpf1F15AcrdonUkV8qrHqMRtPKVXFFS/OXuestWyzTmzGUnx/Ao5dSvUNPWVtxkbTaqn+9kyVRIyHw2dgckLXu2i7nJgZgh4YT/ng03Uue5LMnAOzw5xq03vl+UWIj418vhtIxx+SuljuBSurKptgNBzPLJb3OHLpHz2SiGu8mL6OuZYvxx+rNLMs9rvBC6cQJWMzg7Qseo+5MXvlh2oTvWATWVX1p0RaKAlPJzHG98ac94ArvGfUEoIiFT4cz49qY7APysh4rM+5vA4LGzx2vh452ybzNIeuqGyCQ6p6sVbJy1l5o+mLZXvjfK/KavpiWaFcMDc90w9SYh3VyGWMg0gEpSw9rJGKuCdourb5lyXsC7nXDkxeJMbClzWSAXuMXlVkqvJprX0Zu2BifCIyv76FBy/Y/0Y8AV3Xj4Vsdi1vFnJXb+XTQv7ymp4tJINr+reQfFMj6XJv8MY0kXlWW1PCdVDLpfOpbk9wsZvk5xddbTa/u/Ci7Nk905kss99IVujdcZc9wF6qP0/vO1LJrdo6HrD7dtmHEmruLaid5+8MrYjgwOzl+W6en8LTUp/3qLE4oQ95mro0O0J+rtqiJ7UvfkBPeYqzioz31Lc4J7Xw/aSyeyyLKveVgek7ZdxtIO+g7+9zf7g+bmxuFVuVu0dVx2NoyD8jc0sruH1zlp6HdB29s5F19HpTZ/pJTfrUdgoufZO5qkEs8mE23uL4zkbW/zhEPEkX95l3gVy+qy2HV+TM/Jt13X1//i00xrOPb5+dMTW8+8w6nx3SRa3qm7I+hexN3fklf7dRFZK7w7u3J/RV8SdvkSe8BQ7AUfDo20BGaU0rI867j8aUycypK+XdrbihNio3To03aVtayqvA4O2aDuWarrt6vLbZmvwXa3rwYrvR+q7RfPZg89FD8w/XTXVH3VX34acd9QiVeQC/OurvlfWVL1bubPyx8dfGPxv/atYbK0ZmQ1VeG//9D9ub+9I=</latexit>

Yi (0) , Ti = 0 =) Yi (0) > Yi (1)Contrapositive: Ti = 1 (= Yi (0)  Yi (1)<latexit sha1_base64="q1/SOtUOVSUDn0j/vUHGC7xFOec=">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</latexit>

OTS Bound 1 implication we used: E[Y (1) | T = 0]  E[Y | T = 0]<latexit sha1_base64="Eui1M4nGH2vBrYURJfeCmLq9mbc=">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</latexit>

OTS Bound 2 implication we’ll use: E[Y (1) | T = 0]  E[Y | T = 1]
<latexit sha1_base64="M7DwQ7lnJFzJ8eNGP6ZfH6MVtHo=">AAAUBXicrVhLb9tGEN6kr8R9OdGxFzZKgRSQVclN4VwEBLAdtEUDOIVfrWUYErmSCPEVkrLqCDr31xS9FL32V/TQS/tXOvPtUiT1IKXAEiSuZme+mZ3X7qobOHYUNxr/3Ln7zrvvvf/BvftbH3708Sefbj94eBr5o9CUJ6bv+OF5txNJx/bkSWzHjjwPQtlxu4486w73ef7sWoaR7XvH8U0gL91O37N7ttmJiXS1/V378KLtj2LTd+WT5pdG27Utox0TROxKLzZaRuPSaDvytZFhXMLVvLzarjbqDbyMxUFTD6pCv478B/d/FG1hCV+YYiRcIYUnYho7oiMiel+IpmiIgGiXYkK0kEY25qWYii2SHRGXJI4OUYf03adfF5rq0W/GjCBtkhaHPiFJGuILzWPRuAeqerJ+I8O7SscE2GzjDT27GtMlaiwGRC2TSzjXl+vS2y1ZdUxreIbV2rSSABT2g5lbc4+eDv2OaYX8fUOckkYWSYU0MonmEFVRWEdIT+V59s0AkeiAT9KIrS6yu8hqnvfpPSSsDo0jWMq2GuKFjosHzRK2Mo+DmK5G/IVWqOwrwurN1mAj7moVzHtMlKF4Q6M8cpHObHat5oo18nSJLuaPweERd0QUH5lvk/9s4sjnqkmYrKODSPaxlgAZVNfI32M+yeaA4rkDeyJEzUC+2dAT6MpK7U7sdOjZBXZI8hOaG0AP+6JG9khgh8jexIs1cIf0a4xfJmw05+h1VCbHs0Zr4uyqEa5PM/YMS3lC+TOxSFk3IZqh3zviB+iXFPcavMJ5XSNZHxaqLI6gy9Oxb1FfYb+49M1U9mxCqUGHCzsGyJYh0VJ9Con1fUWriFA/jkad0NOB1wKg1KCd/TPG2MWaOKIj6B7T2IIeljao19XFnrZjmtPKvrHRnRa1Gsg/rl3VGee1spyHSjV0VragqSGe6hVzBNjurNdM3Q1VT46wuumcVV3Yn3YNztG8fSyVxicfKwvZOIBUHpe9uHydy9ayizqpIZbs+z5xtCDV0/QItRloTVuZ+tvXVeIjHiaers4JVe+sbTw3MyG6qz3lEJ+qc96TeOY1YXaQU214LztfhHio5ZPdgDv+BNRiuVNUVl42pNFkNlMsb2Mk4R814ogrvAC+DrCWz2cfIzOzLvbt4HkLiJyxsUZaF91Cbixf+bGoUi+oUlbl/cmRZkqI00lA48czzseEy7xDaPGIwhU+mc1PS2NwQJSp9lgPc0kdpjbY6HwWONs68xSHqqh4joOrulgr5+WyvFH0Yllrlu95WUUvlmXKNXLT1v0gAtZ5iVyMOLCEm8nS4xIpH3uCoiubf1rDPg977UjnRagtPCuRdNFj1Kp8XZUvS+2L0QVD7ROW+fktPHiN/jfFCWhTP6ay8UbeTOVu3sqnqfx4Q8+mku6G/k0l35RISuwN9ozGMq9Ka4q5jkq5VD6V7QmSdpPk/KKqrYbvLnmR92xLdyZD7zecFWp3bKEH1Nbqz4v7Dldys7SOR+i+XfShEJqtgto5uTW0NIIjvZcnu3lyCo8yfd6GxvSEPsFpaqx3hORctQNPKl98Sz3lJZ1VeLwvvqZzUpO+X+R2j3VR+b4y0n0ni7tLyHvU9w+wP2yOG+hbxU7u7pHXcUgaks9U39JSbkefpVchbaJ3ObKKnrVwpp/XpE5tA+JSN5mbEsQ0H5bjFcd3ObL6x8HHSTq9z9wGcvauth5emjOrb9Zl9/3Vt9CAnj36dtAZI817gKxz0CEl1aq6KatTyP7CnZ/zd5eqgnN38ujhnL48/vwtso9b4Ig4Uh51G4ghrWhZxFX30QAysT51Rbi7pTfUeu7GqfDmbYsyeeVqvJbuUFJ33a2r7Wpz/l+sxcHpbr35Tb3x6mn1+TP9D9c98Zl4JJ6Qn/bEc6rMI/KrKX4Xf4t/xX+VXyu/Vf6o/KlY797RMhWRe1X++h/6oupL</latexit>

Proof: E[Y (1) | T = 0] = E[Y (1) | Y (0) > Y (1)]<latexit sha1_base64="EjjEZ8AnvzKw6QZ52nI+ep+IfOA=">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</latexit><latexit sha1_base64="EjjEZ8AnvzKw6QZ52nI+ep+IfOA=">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</latexit><latexit 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 E[Y (1) | Y (0)  Y (1)]= E[Y (1) | T = 1]
= E[Y | T = 1]

Brady Neal

Bounds: Optimal Treatment Selection

27 / 48

OTS Upper Bound 2 Preliminaries
OTS Assumption: Ti = 1 =) Yi (1)<latexit sha1_base64="y9WY9C9eeEOVDI3ZnSJL/kkEjgs=">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</latexit>

Yi (0) , Ti = 0 =) Yi (0) > Yi (1)Contrapositive: Ti = 1 (= Yi (0)  Yi (1)<latexit sha1_base64="q1/SOtUOVSUDn0j/vUHGC7xFOec=">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</latexit>

OTS Bound 1 implication we used: E[Y (1) | T = 0]  E[Y | T = 0]<latexit sha1_base64="Eui1M4nGH2vBrYURJfeCmLq9mbc=">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</latexit>

OTS Bound 2 implication we’ll use: E[Y (1) | T = 0]  E[Y | T = 1]
<latexit sha1_base64="M7DwQ7lnJFzJ8eNGP6ZfH6MVtHo=">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</latexit>

Proof: E[Y (1) | T = 0] = E[Y (1) | Y (0) > Y (1)]<latexit sha1_base64="EjjEZ8AnvzKw6QZ52nI+ep+IfOA=">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</latexit><latexit sha1_base64="EjjEZ8AnvzKw6QZ52nI+ep+IfOA=">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</latexit><latexit sha1_base64="EjjEZ8AnvzKw6QZ52nI+ep+IfOA=">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</latexit><latexit sha1_base64="EjjEZ8AnvzKw6QZ52nI+ep+IfOA=">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</latexit>

 E[Y (1) | Y (0)  Y (1)]= E[Y (1) | T = 1]
= E[Y | T = 1]

Brady Neal

Bounds: Optimal Treatment Selection

27 / 48

OTS Upper Bound 2
OTS assumption tells us that E[Y (1) | T = 0]  E[Y | T = 1]
<latexit sha1_base64="M7DwQ7lnJFzJ8eNGP6ZfH6MVtHo=">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</latexit>

Brady Neal

Bounds: Optimal Treatment Selection

28 / 48

OTS Upper Bound 2
OTS assumption tells us that E[Y (1) | T = 0]  E[Y | T = 1]
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E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1

⇡ E[Y (0) | T = 1]

(1⇡) E[Y | T = 0]

 ⇡ E[Y | T = 1] + (1⇡) E[Y | T = 1]

= E[Y | T = 1]

(1⇡aBrady Neal⇡) E[Y (1) | T = 0]

(1

(Observational-Counterfactual
Decomposition)⇡) E[Y | T = 0]⇡a⇡) E[Y | T = 0]

Bounds: Optimal Treatment Selection

28 / 48

OTS Upper Bound 2
OTS assumption tells us that E[Y (1) | T = 0]  E[Y | T = 1]
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E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

 ⇡ E[Y | T = 1] + (1 ⇡) E[Y | T = 1]
⇡ a (1 ⇡) E[Y | T = 0]
= E[Y | T = 1]

Brady Neal⇡a(1⇡) E[Y | T = 0]

Bounds: Optimal Treatment Selection

28 / 48

OTS Upper Bound 2
OTS assumption tells us that E[Y (1) | T = 0]  E[Y | T = 1]
<latexit 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E[Y (1)

Y (0)] = ⇡ E[Y | T = 1] + (1
⇡ E[Y (0) | T = 1]⇡) E[Y (1) | T = 0]
(Observational-Counterfactual
Decomposition)(1 ⇡) E[Y | T = 0]

 ⇡ E[Y | T = 1] + (1 ⇡) E[Y | T = 1]
⇡ a (1 ⇡) E[Y | T = 0]
= E[Y | T = 1]

Brady Neal⇡a(1⇡) E[Y | T = 0]

Bounds: Optimal Treatment Selection

28 / 48

Question:
Prove a new lower bound using the versionof the OTS assumption that we used in thelast slide.

OTS Complete Bound 2 and Running Example

Brady Neal

Bounds: Optimal Treatment Selection

30 / 48

OTS Complete Bound 2 and Running Example<latexit sha1_base64="szjUafbJenD3debcXC7sO3FQ12g=">AAAVQnicrVhLb9tGEF67r8R9Ja5v6YGN0yBBZVVyUzgXAQFsB22RAE5h59HICChqJRHmK+TKjiPo2H/WP9G/UPRS9NpDZ75diqQepBREgsXl7Mw3s/PaXXciz01Uo/Hn2voHH3708SdXrm58+tnnX3x57frm0yQcxo48cUIvjJ937ER6biBPlKs8+TyKpe13PPmsc7bP88/OZZy4YXCsLiN56tv9wO25jq2I9Or62u/tw5ftcKic0Jd3mnetHWvy1rh7at1ue/K1leOx2r7btdqKlChfBspqWc1Tlopcq12zbBreaer3u0woF22cWu32hsWfKjv6bIdWUmnOd0Ub2Kjl7VDyjRr9HCgZn9ue9UgGfTUYW7dbq6wLNixlbYOdZ7+6tt2oN/CxZgdNM9gW5nMUXr/6q2iLrgiFI4bCF1IEQtHYE7ZI6PtSNEVDREQ7FSOixTRyMS/FWGyQ7JC4JHHYRD2j3z69vTTUgN4ZM4G0Q1o8+otJ0hLfGp4ujXug6ifrt3K8i3SMgM02XtKzYzB9oioxIGqVXMq5vFyHvn7FqhWt4T5W69JKIlDYD05hzT16evSuaIX8e0mckkZdkopp5BDNI6qmsI6Yntrz7JsBImGDT9KIrS6zu8xqng/pe0ZYNo0TWMq2WuKhiUsAzRK2Mo+HmC5GfEMr1PaVYfUma3ARd70K5j0mypl4S6MicpnOfHYt5lIGeTxHF/MrcATEnRAlROa75D+XOIq56hAm67ARyT7WEiGD6gb5F8yn2RxRPHdgT4KoWcg3F3oiU1mZ3amdHj07wI5JfkRzA+hhX9TIHgnsGNmberEG7pjeLvDmwEZnil5HZXI8a7Qmzq4a4YY0406wtCe0P1OLtHUjolnmuyMeQb+kuNfgFc7rGsmGsFBncQJdgYl9i/oK+8WnX6ayZ1NKDTp82DFAtpwRLdOnkVjf97SKBPXjGdQRPT14LQJKDdrZPxcY+1gTR3QI3Rc07kIPS1vU6+piz9gxLmhl37joTrNaLeQf167ujNNaWS5ApVomK1vQ1BD3zIo5Amx33muO6Ya6JydY3XjKqg7sz7oG52jRPpbK4lOMVRfZOIBUEZe9OH+d89ayizqpIZbs+z5xtCDVM/QEtRkZTRu5+ts3VRIiHg6evskJXe+s7WJqZkR033jKIz5d57wn8cxrwrSRU214Lz9fhnho5NPdgDv+CNRyuaeorKJsTKPRZKZc3sVIwj96xBHXeBF8HWEt30z+rNzMstjvBy+YQeSMVQZpWfQucmP+yo/FNvWCbcqqoj850kyJcTqJaHxrwnmLcJn3DFoConCFjybz48oYHBBlbDzWw1xah5kNLjpfF5xtk3maQ1eUmuLgqi7Xynk5L280vVy2O8n3oqyml8sy5Ry56Zp+kADreYWcQhxYws9l6XGFVIg9QdO1zS+WsC/AXjs0eREbC59VSProMXpVoanKx5X2KXTB2PiEZX57Bw+eo/+NcQJa1Y+ZrFrJm5nc5Tv5NJO/WNGzmaS/on8zybcVkhJ7gzuhscyTyppirqNKLp1PVXuCpN0kPb/oaqvht0Ne5D27azqTZfYbzgq9O7bQA2pL9efZfYcruVlZx0N03w76UAzN3ZLaOXlvaFkEh2YvT3fz9BSe5Pq8C43ZCX2E09SF2RHSc9UOPKl98RP1lMd0VuHxvviBzklN+n1Y2D2WReX7ytD0nTzuLiHvUd8/wP6wOm5kbhU7hbtHUcchaUj/xuaWlnF75iy9CGkVvfORdfS6M2f6aU361DYgLn2TuaxAzPJhPl55fOcj6/84hDhJZ/eZ94Gcv6sth5flzOKbddV9f/EtNKJnj349dMbE8B4g6zx0SEm1qm/K+hSyP3Pn5/zdparg3B3d3JzSV8SfvkX2cQscEkfGo28DCtKalkdcdB+NIKPMqSvB3S27odYLN06NN21bkssr3+C1TIeSputuvLq23Zz+L9bs4OluvfljvfHk3vaD++Y/XFfEDXFT3CE/7YkHVJlH5Fdn7e/1zfUb619v/bH119Y/W/9q1vU1I/OVKHy2/vsfiCFK3w==</latexit>

E[Y (1)
E[Y (1)

Y (0)]  E[Y | T = 1]
Y (0)]⇡ E[Y | T = 1] + (1Interval Length = (1Brady Neal

⇡a

(1
⇡) a

⇡) E[Y | T = 0]
E[Y | T = 0]⇡) E[Y | T = 1] + ⇡ E[Y | T = 0]

Bounds: Optimal Treatment Selectiona30 / 48

OTS Complete Bound 2 and Running Example<latexit sha1_base64="szjUafbJenD3debcXC7sO3FQ12g=">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</latexit>

E[Y (1)

Y (0)]  E[Y | T = 1]

E[Y (1)

Y (0)]⇡a⇡ E[Y | T = 1] + (1Interval Length = (1

(1
⇡) a

⇡) E[Y | T = 0]
E[Y | T = 0]⇡) E[Y | T = 1] + ⇡ E[Y | T = 0]aRunning example
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

Brady NealE[Y | T = 1] = .9

<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

Bounds: Optimal Treatment Selection

30 / 48

OTS Complete Bound 2 and Running Example<latexit sha1_base64="szjUafbJenD3debcXC7sO3FQ12g=">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</latexit>

E[Y (1)

Y (0)]  E[Y | T = 1]

E[Y (1)⇡a⇡ E[Y | T = 1] + (1Y (0)]

Interval Length = (1

(1
⇡) a

⇡) E[Y | T = 0]
E[Y | T = 0]⇡) E[Y | T = 1] + ⇡ E[Y | T = 0]aRunning example
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

No-assumptions bound:

Brady Neal<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit><latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

0.17  E[Y (1)Y (0)]  0.83

Bounds: Optimal Treatment Selection

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OTS Complete Bound 2 and Running Example<latexit sha1_base64="szjUafbJenD3debcXC7sO3FQ12g=">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</latexit>

E[Y (1)

Y (0)]  E[Y | T = 1]

E[Y (1)⇡a⇡ E[Y | T = 1] + (1Y (0)]

Interval Length = (1

(1
⇡) a

⇡) E[Y | T = 0]
E[Y | T = 0]⇡) E[Y | T = 1] + ⇡ E[Y | T = 0]aRunning example
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">AAATw3icrVjbbttGEN2kt8S9OdFjX9goBfogq5KTwnkREMB2UBQN4BS24zQOAl0oixBvJSmrjqCv6Gv7Yf2bnjm7FEndKAWWIHE1O3Nmdm67q07oOnHSaPx35+4nn372+Rf37u98+dXX33y7++DheRyMoq591g3cILrotGPbdXz7LHES174II7vtdVz7dWd4KPOvr+0odgL/NLkJ7Xde+8p3+k63nYD05jJ0rJbVqD95v1tt1Bt8WYuDphlUlXmdBA/u/64uVU8FqqtGylO28lWCsavaKsb7rWqqhgpBe6cmoEUYOZy31VTtQHYELhscbVCH+L7Cr7eG6uO3YMaU7kKLi08ESUv9YHh6GPdJ1U/Rb+V4V+mYEFtsvMGzYzA9UBM1ALVMLuXcXK6Dt1ey6gRreMbVOlhJSIr4oVtYcx9PF78TrFC+b8BpY9SDVIRRFzQXVE0RHRGe2vPimwEj0SafjZFYvc7udVbLfID3EFhtjGNaKrZa6oWJi0/NNm0VHpcxXY34F1ao7VuH1Z+twWHc9SqE9xSUofqAURF5nc58dq3mSgzydIku4U/I4YM7BiVg5jvwnwOOYq52gSk62ozkFdcSMoPqBvlXzqfZHCKee7QnZtQs5ptDPaGprMzu1E4Xzw6xI8hPMDegHvFFDfbYxI6YvakXa+SO8GvMX13a2J2j11mZEs8a1iTZVQNugBlnhqU9of2ZWqStm4Bmmfee+o36bcS9Rq9IXtcgG9BCncUxdfkm9i30FfGLh2+himdTSo06PNoxYLYMQcv0aSTR9xNWEbN+XIM6wdOl10Ki1Khd/DPm2OOaJKIj6h5j3KMekbbQ6+rqwNgxLWgV3zjsTotaLeaf1K7ujPNaRc5npVomK1vU1FBPzYolAmJ33mtd0w11T465uumcVR3an3UNydGifSKVxacYqx6zcUCpIq54cfk6l61ln3VSYyzF91fgaFGqb+gxazM0mnZy9XdoqiRgPLp8eiYndL2LtvHczAR0z3jKBZ+uc9mTZOZPYLaZU5f0Xn5+HeKxkU93A+n4E1LXy52zsoqyEUaT2cx6eYcjm/7RI4m4xgvp65Br+X72sXIzm2LfDp6/gCgZmxikTdF7zI3lKz9VVfSCKrKq6E+JtFAink5CjB/POB8DV3iH1OKDIhU+mc1PS2NwBMrUeKzPubQOMxscdr4eOS9N5mkOXVHJHIdU9XqtkpfL8kbT18v2ZvlelNX09bJCuWZuOqYfxMS6KJFLGAeR8HJZeloiFXBP0HRt85sN7PO5145MXkTGwtclkh57jF5VYKryZal9CbtgZHwiMn98hAev2f+mPAFt68dMNtnKm5nczUf5NJMfb+nZTNLb0r+Z5IcSSZt7gzOjicyr0poSrpNSLp1PZXuCjd0kPb/oaqvxuwMvyp7dM53JMvuNZIXeHVvsAbWN+vPiviOV3Cyt4xG7b4d9KKLm3praObs1tCyCI7OXp7t5egqPc33eocbshD7haWpsdoT0XLVHT2pf/IKe8hJnFRkfqic4JzXx/aKwe2yKKveVkek7edx9IB+g7x9xf9geNzS3ir3C3aOo4xga0s/U3NIybtecpVchbaN3ObKOXm/hTD+vSZ/aBuDSN5mbEsQsH5bjrY/vcmT9j0PAk3R2n7kN5PxdbTO8LGdW36zL7vurb6Ehnn18u+yMseE9Yta57JA2alXflPUp5HDhzi/5u4+qkNydPHo4p6+IP3+LvOItcASOjEffBhJKa1oecdV9NKRMYk5dMe9u2Q21Xrhxarx52+JcXnkGr2U6lG267s773Wpz/l+sxcH5fr35c73x6mn1+TPzD9c99Z16pH6Enw7Uc1TmCfwqN66/1T/q38pxZViJKolmvXvHyFRU4VWZ/g8Zn9C0</latexit>

E[Y | T = 1] = .9

No-assumptions bound:
OTS Bound 2:<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

0.17  E[Y (1)Y (0)]  0.83

0.07  E[Y (1)Y (0)]  0.76

<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit><latexit sha1_base64="KjmpZmg5NNffP+QIED+hYkksN24=">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</latexit>

Interval Length = 0.69

Brady Neal

Bounds: Optimal Treatment Selection

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OTS Complete Bound 2 and Running Example<latexit sha1_base64="szjUafbJenD3debcXC7sO3FQ12g=">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</latexit>

E[Y (1)

Y (0)]  E[Y | T = 1]

E[Y (1)⇡a⇡ E[Y | T = 1] + (1Y (0)]

Interval Length = (1

(1
⇡) a

⇡) E[Y | T = 0]
E[Y | T = 0]⇡) E[Y | T = 1] + ⇡ E[Y | T = 0]aRunning example
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

No-assumptions bound:
OTS Bound 2:<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

0.17  E[Y (1)Y (0)]  0.83

0.07  E[Y (1)Y (0)]  0.76

<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit><latexit sha1_base64="KjmpZmg5NNffP+QIED+hYkksN24=">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</latexit>

Interval Length = 0.69

Brady Neal

Bounds: Optimal Treatment Selection

Identified the signof the effect!
30 / 48

Comparing and Mixing OTS Bounds
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

Brady NealE[Y | T = 1] = .9

<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

Bounds: Optimal Treatment Selection

31 / 48

Comparing and Mixing OTS Bounds
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

No-assumptions bound:

Brady Neal<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit><latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

0.17  E[Y (1)Y (0)]  0.83

Bounds: Optimal Treatment Selection

31 / 48

Comparing and Mixing OTS Bounds
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

No-assumptions bound: 0.17  E[Y (1) Y (0)]  0.83
OTS Bound 1: 0.14  E[Y (1) Y (0)]  0.27
<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit><latexit sha1_base64="XMUf2rSbtZk24I7/igjQCRTQueM=">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</latexit>

Interval Length = 0.41

Brady Neal

Bounds: Optimal Treatment Selection

31 / 48

Comparing and Mixing OTS Bounds
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

No-assumptions bound: 0.17  E[Y (1) Y (0)]  0.83
OTS Bound 1: 0.14  E[Y (1) Y (0)]  0.27
<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit><latexit sha1_base64="XMUf2rSbtZk24I7/igjQCRTQueM=">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</latexit>

Interval Length = 0.41

OTS Bound 2:

0.07  E[Y (1)<latexit sha1_base64="KjmpZmg5NNffP+QIED+hYkksN24=">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</latexit>

Y (0)]  0.76

Interval Length = 0.69

Brady Neal

Bounds: Optimal Treatment Selection

31 / 48

Comparing and Mixing OTS Bounds
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

No-assumptions bound: 0.17  E[Y (1) Y (0)]  0.83
OTS Bound 1: 0.14  E[Y (1) Y (0)]  0.27
<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit><latexit sha1_base64="XMUf2rSbtZk24I7/igjQCRTQueM=">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</latexit>

Interval Length = 0.41

OTS Bound 2:

0.07  E[Y (1)<latexit sha1_base64="KjmpZmg5NNffP+QIED+hYkksN24=">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</latexit>

Y (0)]  0.76

Interval Length = 0.69

Brady Neal

Identified the sign of the effect,
but gives a 68% larger interval

Bounds: Optimal Treatment Selection

31 / 48

Comparing and Mixing OTS Bounds
Potential outcomes bounded between 0 (a) and 1 (b)<latexit sha1_base64="0qNgQeOfCCNVWbvIY5PG57G3lcA=">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</latexit>⇡ = 0.3<latexit sha1_base64="7Rp8/HTQ6RDjGzfT4AmsnMitKdc=">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</latexit>

E[Y | T = 1] = .9

<latexit sha1_base64="xW9LuNHNXw/HCg+T3VfX5m8sUYE=">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</latexit>

E[Y | T = 0] = .2

No-assumptions bound: 0.17  E[Y (1) Y (0)]  0.83
OTS Bound 1: 0.14  E[Y (1) Y (0)]  0.27
<latexit sha1_base64="TdWecIwojCY7QlkdYZZF/zpuSfI=">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</latexit><latexit sha1_base64="XMUf2rSbtZk24I7/igjQCRTQueM=">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</latexit>

Interval Length = 0.41

OTS Bound 2:

0.07  E[Y (1)<latexit sha1_base64="KjmpZmg5NNffP+QIED+hYkksN24=">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</latexit>

Y (0)]  0.76

Interval Length = 0.69

Identified the sign of the effect,
but gives a 68% larger interval

OTS Upper Bound 1 and OTS Lower Bound 2:
0.07  E[Y (1)<latexit sha1_base64="f7tn3SKr4KzEoKLN6lBNQd0rxxk=">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</latexit>

Brady Neal

Y (0)]  0.27

Interval Length = 0.2<latexit sha1_base64="JtD96c4+VAUmHGKsRi0gW1ohzfM=">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</latexit>

Bounds: Optimal Treatment Selection

31 / 48

Bounds
No-Assumptions Bound
Monotone Treatment Response
Monotone Treatment Selection
Optimal Treatment Selection

Sensitivity Analysis
Linear Single Confounder
Towards More General Settings

Brady Neal

Sensitivity Analysis

32 / 48

Unobserved Confounder Setting<latexit sha1_base64="eRljzOoTMx+Fb5CMXgiXG590eak=">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</latexit>

W

T

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Brady NealY

Y (0)] = EW [E[Y | T = 1, W ]Sensitivity Analysis

E[Y | T = 0, W ]]33 / 48

Unobserved Confounder Setting
Last section, we completely threw out the unconfoundedness assumption.
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W

T

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Brady NealY

Y (0)] = EW [E[Y | T = 1, W ]Sensitivity Analysis

E[Y | T = 0, W ]]33 / 48

Unobserved Confounder Setting
Last section, we completely threw out the unconfoundedness assumption.
Now, we assume the observed W and the unobserved U gives us
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Brady Neal

W

U

T

Y

WW,
] UE[Y|T =W0,]] W, U ]]Y (0)] = EW,U
[E[Y| T| T==1,1,] E[Y| T0,=
W [E[Y
Sensitivity Analysis

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Unobserved Confounder Setting
Last section, we completely threw out the unconfoundedness assumption.
Now, we assume the observed W and the unobserved U gives us
unconfoundedness<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">AAAWTHicrVjbbttGEN3YvSTuLUnhvrQPbOwUCSCrstsiAQIDAZwERdEUSeFcWssIKHIlEeKt5MqqI+gb+md9738UfSkKdObsUiQlipSDSLC4nJ05Mzu33XUv9r1UdTp/XdrYfOfd996/fGXrgw8/+viTq9euP0+jceLIZ07kR8nLnp1K3wvlM+UpX76ME2kHPV++6I2OeP7FmUxSLwqP1XksTwN7EHp9z7EVkV5d2/ij25MDL5wqb/Q69hw1TuRsyzKfm9ZPkSvT+Xs3pNeTVNlKnlq3uk4U9qNx6BL+bctW1q1Oq3Pbmu4WJ3Zn96rEW66dDqXLKOMw6qUyOZNuCe8k8QZDZR1GfauId8r4lSIrNLEKRS5RgQwVwfakH02qYedsNVDRWDlRIMtAlfYA0rBngAXPPnQHRc/GthouulQSS8n6e2vxZyYuc69w9Rp66iVXayx4vpK3K0O3lHuvru502h18rOXBvhnsCPN5El278rPoCldEwhFjEQgpQqFo7AtbpPQ9EfuiI2KinYop0RIaeZiXYia2SHZMXJI4bKKO6HdAbyeGGtI7Y6aQdkiLT38JSVripuFxadwHVT9Zv1XgXaVjCmy28ZyePYMZEFWJIVGb5DLO9eV69A0aVq1oDXexWo9WEoPCfnBKa+7T06d3RSvk33PilDRySSqhkUM0n6iawjoSemrPs2+GiIQNPkkjtrrO7jqreT6i74iwbBqnsJRttcQjE5cQmiVsZR4fMV2N+DutUNtXh9Wfr8FD3PUqmPeYKCPxmkZl5DqdxexazaUM8qxCF/MrcITEnRIlQuZ75D+POMq56hAm67ARyQHWEiOD2gb5B8xn2RxTPPdgT4qoWcg3D3piU1m53ZmdPj17wE5IfkpzQ+hhX7TIHgnsBNmbebEF7oTeJnhzYKOzQG+jMjmeLVoTZ1eLcCOa8eZY2hPan5lF2rop0Szz3RM/Qr+kuLfgFc7rFslGsFBncQpdoYn9IfUV9ktAv0xlz2aUFnQEsGOIbBkRLdenkVjf17SKFPXjG9QpPX14LQZKC9rZPxOMA6yJIzqG7gmNXehhaYt6XVvcMXbMSlrZNx6607JWC/nHtas746JWlgtRqZbJykNo6ohvzYo5Amx30WuO6Ya6J6dY3WzBqh7sz7sG52jZPpbK41OOlYtsHEKqjMterF5n1VoOUCctxJJ9PyCOQ0j1DT1FbcZG01ah/o5MlUSIh4NnYHJC1ztrmyzMTIkeGE/5xKfrnPcknvmNMG3kVBfeK87XIT408tluwB1/Cmq93HNUVlk2odF0PlMv72Ek4R894ohrvBi+jrGWL+d/VmFmXey3gxcuIXLGKoO0LrqL3Khe+bHYoV6wQ1lV9idHmikJTicxjXfnnLuEy7wjaAmJwhU+nc/PGmPwgCgz47E+5rI6zG3w0PlccHZN5mkOXVFqgYOrul4r52VV3mh6vaw7z/eyrKbXyzLlDLnpmX6QAutlg5xCHFgiKGTpcYNUhD1B07XNv6xhX4i9dmzyIjEWvmiQDNBj9KoiU5WPG+1T6IKJ8QnL/PoGHjxD/5vhBHRRP+ay6kLezOXO38inufzkgp7NJYML+jeXfN0gKbE3eHMayzxtrCnmetLIpfOpaU+QtJtk5xddbS389siLvGe7pjNZZr/hrNC74yF6QGut/ry873Al7zfW8Rjdt4c+lECzW1M7z94aWh7BsdnLs908O4WnhT7vQWN+Qp/iNDUxO0J2rtqDJ7Uvvqee8pjOKjw+Et/QOWmffh+Vdo91Ufm+MjZ9p4h7QMh3qO8/wP5wcdzY3Cr2SnePso6HpCH7m5lbWs7tm7P0KqSL6K1G1tFzl870i5r0qW1IXPomc96AmOdDNV59fKuR9X8cIpyk8/vM20Au3tXWw8tzZvXNuum+v/oWGtOzT78+OmNqeB8g63x0SEm1qm/K+hRytHTn5/w9oKrg3J3euL6gr4y/eIsc4BY4Jo6cR98GFKQ1rYi46j4aQ0aZU1eKu1t+Q22Xbpwab9G2tJBXgcE7NB1Kmq679erqzv7if7GWB88P2vvftTtPD3bu3zX/4bosPhc3xC3y0x1xnyrzCfnV2fhn84vNm5tfbf+5/ff2v9v/adaNS0bmU1H6fPbe/3rTp4E=</latexit>

E[Y (1)<latexit sha1_base64="qEU0nKhHJfNQr73O6QeY2sLiLXE=">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</latexit><latexit sha1_base64="7h4SIYwEFBbP5TqSm2XIubJIUPU=">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</latexit>

W

U

T

Y

WW,
] UE[Y|T =W0,]] W, U ]]Y (0)] = EW,U
[E[Y| T| T==1,1,] E[Y| T0,=
W [E[Y<latexit sha1_base64="vei1mn9NO6vHWjVzBryobX//Eg0=">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</latexit>?

⇡ EW [E[Y | T = 1, W ]Brady Neal

Sensitivity Analysis

E[Y | T = 0, W ]]33 / 48

Bounds
No-Assumptions Bound
Monotone Treatment Response
Monotone Treatment Selection
Optimal Treatment Selection

Sensitivity Analysis
Linear Single Confounder
Towards More General Settings

Brady Neal

Sensitivity Analysis: Linear Single Confounder

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Linear SCM<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">AAAWTHicrVjbbttGEN3YvSTuLUnhvrQPbOwUCSCrstsiAQIDAZwERdEUSeFcWssIKHIlEeKt5MqqI+gb+md9738UfSkKdObsUiQlipSDSLC4nJ05Mzu33XUv9r1UdTp/XdrYfOfd996/fGXrgw8/+viTq9euP0+jceLIZ07kR8nLnp1K3wvlM+UpX76ME2kHPV++6I2OeP7FmUxSLwqP1XksTwN7EHp9z7EVkV5d2/ij25MDL5wqb/Q69hw1TuRsyzKfm9ZPkSvT+Xs3pNeTVNlKnlq3uk4U9qNx6BL+bctW1q1Oq3Pbmu4WJ3Zn96rEW66dDqXLKOMw6qUyOZNuCe8k8QZDZR1GfauId8r4lSIrNLEKRS5RgQwVwfakH02qYedsNVDRWDlRIMtAlfYA0rBngAXPPnQHRc/GthouulQSS8n6e2vxZyYuc69w9Rp66iVXayx4vpK3K0O3lHuvru502h18rOXBvhnsCPN5El278rPoCldEwhFjEQgpQqFo7AtbpPQ9EfuiI2KinYop0RIaeZiXYia2SHZMXJI4bKKO6HdAbyeGGtI7Y6aQdkiLT38JSVripuFxadwHVT9Zv1XgXaVjCmy28ZyePYMZEFWJIVGb5DLO9eV69A0aVq1oDXexWo9WEoPCfnBKa+7T06d3RSvk33PilDRySSqhkUM0n6iawjoSemrPs2+GiIQNPkkjtrrO7jqreT6i74iwbBqnsJRttcQjE5cQmiVsZR4fMV2N+DutUNtXh9Wfr8FD3PUqmPeYKCPxmkZl5DqdxexazaUM8qxCF/MrcITEnRIlQuZ75D+POMq56hAm67ARyQHWEiOD2gb5B8xn2RxTPPdgT4qoWcg3D3piU1m53ZmdPj17wE5IfkpzQ+hhX7TIHgnsBNmbebEF7oTeJnhzYKOzQG+jMjmeLVoTZ1eLcCOa8eZY2hPan5lF2rop0Szz3RM/Qr+kuLfgFc7rFslGsFBncQpdoYn9IfUV9ktAv0xlz2aUFnQEsGOIbBkRLdenkVjf17SKFPXjG9QpPX14LQZKC9rZPxOMA6yJIzqG7gmNXehhaYt6XVvcMXbMSlrZNx6607JWC/nHtas746JWlgtRqZbJykNo6ohvzYo5Amx30WuO6Ya6J6dY3WzBqh7sz7sG52jZPpbK41OOlYtsHEKqjMterF5n1VoOUCctxJJ9PyCOQ0j1DT1FbcZG01ah/o5MlUSIh4NnYHJC1ztrmyzMTIkeGE/5xKfrnPcknvmNMG3kVBfeK87XIT408tluwB1/Cmq93HNUVlk2odF0PlMv72Ek4R894ohrvBi+jrGWL+d/VmFmXey3gxcuIXLGKoO0LrqL3Khe+bHYoV6wQ1lV9idHmikJTicxjXfnnLuEy7wjaAmJwhU+nc/PGmPwgCgz47E+5rI6zG3w0PlccHZN5mkOXVFqgYOrul4r52VV3mh6vaw7z/eyrKbXyzLlDLnpmX6QAutlg5xCHFgiKGTpcYNUhD1B07XNv6xhX4i9dmzyIjEWvmiQDNBj9KoiU5WPG+1T6IKJ8QnL/PoGHjxD/5vhBHRRP+ay6kLezOXO38inufzkgp7NJYML+jeXfN0gKbE3eHMayzxtrCnmetLIpfOpaU+QtJtk5xddbS389siLvGe7pjNZZr/hrNC74yF6QGut/ry873Al7zfW8Rjdt4c+lECzW1M7z94aWh7BsdnLs908O4WnhT7vQWN+Qp/iNDUxO0J2rtqDJ7Uvvqee8pjOKjw+Et/QOWmffh+Vdo91Ufm+MjZ9p4h7QMh3qO8/wP5wcdzY3Cr2SnePso6HpCH7m5lbWs7tm7P0KqSL6K1G1tFzl870i5r0qW1IXPomc96AmOdDNV59fKuR9X8cIpyk8/vM20Au3tXWw8tzZvXNuum+v/oWGtOzT78+OmNqeB8g63x0SEm1qm/K+hRytHTn5/w9oKrg3J3euL6gr4y/eIsc4BY4Jo6cR98GFKQ1rYi46j4aQ0aZU1eKu1t+Q22Xbpwab9G2tJBXgcE7NB1Kmq679erqzv7if7GWB88P2vvftTtPD3bu3zX/4bosPhc3xC3y0x1xnyrzCfnV2fhn84vNm5tfbf+5/ff2v9v/adaNS0bmU1H6fPbe/3rTp4E=</latexit>

Brady Neal

W

U

T

Y

Sensitivity Analysis: Linear Single Confounder

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Linear SCM<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit><latexit sha1_base64="oLNh5jgGL8lSG3RxxH8XLmVD2Fg=">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</latexit>

W

U

T

Y

T := ↵w W + ↵u U

Y :=Brady Neal

wW +

uU +

T

Sensitivity Analysis: Linear Single Confounder

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Linear SCM<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit><latexit sha1_base64="oLNh5jgGL8lSG3RxxH8XLmVD2Fg=">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</latexit>

W

U

T

Y

T := ↵w W + ↵u U

Y :=

wW +

uU +

Goal: recover
Brady Neal

T<latexit sha1_base64="4kLgsB3V/MeIDLI24ttEC6CioZs=">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</latexit>

Sensitivity Analysis: Linear Single Confounder

35 / 48

Bias in Simple Linear Setting<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit><latexit sha1_base64="oLNh5jgGL8lSG3RxxH8XLmVD2Fg=">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</latexit>

W

U

T

Y

T := ↵w W + ↵u U

Y :=Brady Neal

wW +

uU +

T

Sensitivity Analysis: Linear Single Confounder

36 / 48

Bias in Simple Linear Setting<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit><latexit sha1_base64="oLNh5jgGL8lSG3RxxH8XLmVD2Fg=">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</latexit>

W

U

T

Y

T := ↵w W + ↵u U

Y :=<latexit sha1_base64="ECX4aNrFkygg1eWzyV7O7iEbFzI=">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</latexit>wW +E[Y (1)

Brady Neal

uU +

T

Y (0)] = EW,U [E[Y | T = 1, W, U ]E[Y | T = 0, W, U ]] =Sensitivity Analysis: Linear Single Confounder

36 / 48

Bias in Simple Linear Setting<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit><latexit sha1_base64="oLNh5jgGL8lSG3RxxH8XLmVD2Fg=">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</latexit>

W

U

T

Y

T := ↵w W + ↵u U

Y :=<latexit sha1_base64="ECX4aNrFkygg1eWzyV7O7iEbFzI=">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</latexit>wW +E[Y (1)uU +T

Y (0)] = EW,U [E[Y | T = 1, W, U ]E[Y | T = 0, W, U ]] =<latexit 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EW [E[Y | T = 1, W ]Brady Neal?

E[Y | T = 0, W ]] =Sensitivity Analysis: Linear Single Confounder

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W

U

T

Y

T := ↵w W + ↵u U

Y :=<latexit sha1_base64="ECX4aNrFkygg1eWzyV7O7iEbFzI=">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</latexit>wW +E[Y (1)uU +T

Y (0)] = EW,U [E[Y | T = 1, W, U ]<latexit sha1_base64="zWr5HNn3VSEmW3hoOPrylTWI2uA=">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</latexit>

E[Y | T = 0, W, U ]] =<latexit sha1_base64="bCgkyUx3yWUkqyiD0hPOxiEVoeE=">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</latexit><latexit sha1_base64="bCgkyUx3yWUkqyiD0hPOxiEVoeE=">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</latexit>

EW [E[Y | T = 1, W ]Brady Neal?

E[Y | T = 0, W ]] = +Sensitivity Analysis: Linear Single Confounderu↵u

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Bias in Simple Linear Setting<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit>

W<latexit sha1_base64="oLNh5jgGL8lSG3RxxH8XLmVD2Fg=">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</latexit>

U

T := ↵w W + ↵u U

Y :=

wW +

uU +

Proof coming
after next slideTT<latexit sha1_base64="ECX4aNrFkygg1eWzyV7O7iEbFzI=">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</latexit>

E[Y (1)Y

Y (0)] = EW,U [E[Y | T = 1, W, U ]<latexit sha1_base64="zWr5HNn3VSEmW3hoOPrylTWI2uA=">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</latexit>

E[Y | T = 0, W, U ]] =<latexit sha1_base64="bCgkyUx3yWUkqyiD0hPOxiEVoeE=">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</latexit><latexit sha1_base64="bCgkyUx3yWUkqyiD0hPOxiEVoeE=">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</latexit>

EW [E[Y | T = 1, W ]Brady Neal?

E[Y | T = 0, W ]] = +Sensitivity Analysis: Linear Single Confounderu↵u

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Bias in Simple Linear Setting<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit>

W<latexit sha1_base64="oLNh5jgGL8lSG3RxxH8XLmVD2Fg=">AAAUrXicrVhZb9tGEF6lV+JeTuq3vrBRWhSorEhuCgcFBASwExRFAziFrzZKDR4riRCvkpRVR9Br+xv7M/oPOvPtUiR1UQosQeJqduab2bl2V1bkuUnaav1bu/Pe+x98+NHdezsff/LpZ5/v3n9wnoSj2JZnduiF8aVlJtJzA3mWuqknL6NYmr7lyQtreMTzF9cyTtwwOE1vIvnGN/uB23NtMyXS1e5/3ZTYU18GqfHNjx2ja3rRwLzq2mHQC0eBQ6LXpmcUfxvfzbhGQWglMr6Wzjz/0hnD6HZ3DHp1w1Fqh75UKi2ZVmtUTNsqJEFHeqlp5Mu82q23mi28jMVBWw/qQr9Owvv3fhVd4YhQ2GIkfCFFIFIae8IUCb1fi7ZoiYhob8SEaDGNXMxLMRU7JDsiLkkcJlGH9N2nX681NaDfjJlA2iYtHn1ikjTE15rHoXEPVPVk/UaBd5WOCbDZxht6WhrTJ2oqBkStkss4N5ez6O1XrDqlNTzFal1aSQQK+8EurblHT49+p7RC/r4hTkkjh6RiGtlE84iqKKwjpqfyPPtmgEiY4JM0YqvX2b3Oap4P6T0kLJPGCSxlWw3xQsclgGYJW5nHQ0xXI/5FK1T2rcPqzdbgIu5qFcx7SpSheEujMvI6ncXsWs2VauTpEl3Mn4IjIO6EKCEy3yX/ucRRzlWbMFmHiUj2sZYIGdTUyD9jPsvmiOK5D3sSRM1AvrnQE+nKyu3O7PToaQE7JvkJzQ2gh33RIHsksGNkb+bFBrhj+jXGLxs22nP0JiqT49mgNXF2NQg3pBl3hqU8ofyZWaSsmxDN0O998Qv0S4p7A17hvG6QbAgLVRYn0BXo2Heor7BffPpmKns2ozSgw4cdA2TLkGi5PoXE+h7TKhLUj6dRJ/T04LUIKA1oZ/+MMfaxJo7oCLrHNHagh6UN6nVNcajtmJa0sm9cdKdFrQbyj2tXdcZ5rSwXoFINnZUdaGqJJ3rFHAG2u+g1W3dD1ZMTrG46Z5UF+/OuwTlato+l8viUY+UgGweQKuOyF5evc9laDlAnDcSSfd8njg6kepqeoDYjrWmnUH9HukpCxMPG09c5oeqdtY3nZiZE97WnPOJTdc57Es/8SZgmcqoL7xXn1yE+1/LZbsAdfwLqerlzVFZZNqbRZDazXt7FSMI/asQRV3gRfB1hLV/NPkZhZlPs28ELFhA5Y1ONtCm6g9xYvvJTUadeUKesKvuTI82UGKeTiMaPZpyPCJd5h9ASEIUrfDKbn1bG4JgoU+2xHuayOsxtcNH5HHB2deYpDlVR6RwHV/V6rZyXy/JG0dfLOrN8L8sq+npZplwjN13dDxJgXVbIpYgDS/iFLD2tkAqxJyi6svm3DewLsNeOdF7E2sKLCkkfPUatKtRV+bLSvhRdMNY+YZnf38GD1+h/U5yAtvVjLptu5c1c7uadfJrLj7f0bC7pb+nfXPJthaTE3uDOaCzzqrKmmOukkkvlU9WeIGk3yc4vqtoa+LbIi7xnO7ozGXq/4axQu2MHPaCxUX9e3He4ktuVdTxC97XQh2JodtbUztmtoeURHOm9PNvNs1N4UujzLjTmJ/QJTlNjvSNk56p9eFL54ifqKS/prMLjI/E9nZPa9P2itHtsisr3lZHuO0XcA0I+pL5/jP1he9xI3yr2S3ePso7npCH7TPUtLef29Fl6FdI2epcjq+g5C2f6eU3q1DYgLnWTualAzPNhOd76+C5HVv84hDhJ5/eZ20Au3tU2w8tzZvXNuuq+v/oWGtGzR98eOmOieY+RdR46pKRaVTdldQo5Wrjzc/4eUFVw7k4ePpjTV8afv0X2cQscEUfOo24DKaQVrYi46j4aQSbVp64Ed7f8htos3TgV3rxtSSGvfI3X0R1K6q67c7Vbb8//i7U4OD9otn9otl49qT97qv/huiu+FA/Ft+SnQ/GMKvOE/GrXLmrT2t+1f/Ye753tdff+UKx3alrmC1F67fX/B4d+KkU=</latexit>

U

T := ↵w W + ↵u U

Y :=

wW +

uU +

Proof coming
after next slideTT<latexit sha1_base64="ECX4aNrFkygg1eWzyV7O7iEbFzI=">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</latexit>

E[Y (1)Y

Y (0)] = EW,U [E[Y | T = 1, W, U ]<latexit sha1_base64="zWr5HNn3VSEmW3hoOPrylTWI2uA=">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</latexit>

E[Y | T = 0, W, U ]] =?

<latexit sha1_base64="bCgkyUx3yWUkqyiD0hPOxiEVoeE=">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</latexit><latexit sha1_base64="bCgkyUx3yWUkqyiD0hPOxiEVoeE=">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</latexit>

EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] = +u↵u<latexit sha1_base64="TLdb9LB2IawHGV9IHLw7pLDpis0=">AAAU9XicrVhZb9tGEF6lV+IeOeq3vrBxChSorEpuiuRFQFAnQVE0QFI4R2saBkWuLMK8Sq6sOIR+StGXoq/9FX1tn/tvOvPtSiR1kHJgCRJXszPfzM61uxokgZ+pbve/1pV33n3v/Q+uXtv68KOPP7l+4+atF1k8Tl353I2DOH01cDIZ+JF8rnwVyFdJKp1wEMiXg9N9nn95JtPMj6MDdZ7Io9A5ifyh7zqKSMc3W/dtJV+r/Dvfyax4aE0t+9FxbrtxNIzHkUeSRAnkUB3SxKEdj5Ubh9KyQ9+zbEWaVCgjZfWtXtsqSx1Zuw0C3UUBO/VPRuqIpmxPBsqxvrLsYeq4uT2Qyjm2x1E8yGR6Jr2S2JkTTHPbCZJRHQcbozH7l4Z5fGOn2+niZS0PemawI8zraXzz2k/CFp6IhSvGIhRSRELROBCOyOh9KHqiKxKiHYmcaCmNfMxLMRVbJDsmLkkcDlFP6fuEfh0aakS/GTODtEtaAvqkJGmJLwyPR+MhqPrJ+q0S7zodObDZxnN6DgxmSFQlRkRtkptxbi43oHfYsGpFa7iP1fq0kgQU9oNbWfOQngH9VrRC/j4nTkkjj6RSGrlEC4iqKawjpaf2PPtmhEg44JM0Yqvr7K6zmudjep8SlkPjDJayrZZ4bOISQbOErcwTIKbrEV/TCrV9dVjD+Rp8xF2vgnkPiHIq3tCoilyns5xd67mUQZ6u0MX8ChwRcWdEiZH5PvnPJ45qrrqEyTocRPIEa0mQQR2D/APmZ9mcUDx3YU+GqFnINx96ElNZhd0zOwN6DoCdknxOcyPoYV+0yR4J7BTZO/NiG9wp/Zrglwsb3QV6B5XJ8WzTmji72oQb04w/x9Ke0P6cWaSty4lmmfeu+BH6JcW9Da9wXrdJNoaFOosz6IpM7PvUV9gvIX0zlT07o7ShI4QdI2TLKdEKfRqJ9X1Nq8hQP4FBzekZwGsJUNrQzv6ZYBxiTRzRMXRPaOxBD0tb1Os64p6xY1rRyr7x0Z2WtVrIP65d3RkXtbJchEq1TFb2oakr7poVcwTY7rLXXNMNdU/OsLrpglUD2F90Dc7Rqn0sVcSnGisP2TiCVBWXvbh6navWsoc6aSOW7PsT4uhDamjoGWozMZq2SvW3b6okRjxcPEOTE7reWdtkYSYnemg8FRCfrnPek3jmV8J0kFM2vFeer0N8ZORnuwF3/BzUerkXqKyqbEqjfD5TL+9jJOEfPeKIa7wEvk6wls/nH6s0syn25eBFS4icscogbYruITdWr/xA7FAv2KGsqvqTI82UFKeThMZ35px3CJd5T6ElIgpXeD6fnzbG4CFRpsZjQ8zN6rCwwUfn88Bpm8zTHLqi1AIHV3W9Vs7LVXmj6fWy3jzfq7KaXi/LlDPkpm/6QQasVw1yCnFgibCUpQcNUjH2BE3XNv+8gX0R9tqxyYvUWPiyQTJEj9Grik1VPmm0T6ELpsYnLPPLW3jwDP1vihPQRf1YyKoLebOQO38rnxbykwt6tpAML+jfQvJNg6TE3uDPaSzzrLGmmOtpI5fOp6Y9QdJuMju/6Gpr43tAXuQ92zOdyTL7DWeF3h376AHtjfrz8r7DldxrrOMxuu8AfSiFZq+mdp5fGloRwbHZy2e7+ewUnpX6vA+NxQk9x2lqYnaE2blqF57UvvieesoTOqvweF98Q+ekHn0/ruwem6LyfWVs+k4Zd4+Q71Hff4j94eK4iblV7FbuHlUdj0jD7DM1t7SCOzBn6XVIF9G7GllHz1s60y9q0qe2EXHpm8x5A2KRD6vx6uO7Gln/4xDjJF3cZy4DuXxX2wyvyJn1N+um+/76W2hCzyF9B+iMmeF9iKwL0CEl1aq+KetTyP7SnZ/zd4+qgnM3v31rQV8Vf/EWeYJb4Jg4Ch59G1CQ1rQy4rr7aAIZZU5dGe5uxQ21U7lxarxF27JSXoUGr286lDRdd+v4xk5v8V+s5cGLvU7v20732d2dB/fNP1xXxWfitviS/HRPPKDKfEp+dVu/t/5u/dP6d3uy/dv2H9t/atYrLSPzqai8tv/6H+nlR3M=</latexit>

Bias of EW [E[Y | T = 1, W ]Brady NealE[Y | T = 0, W ]] = +Sensitivity Analysis: Linear Single Confounder

u

↵u

=

u

↵u
36 / 48

Bias in Simple Linear Setting<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">AAAWTHicrVjbbttGEN3YvSTuLUnhvrQPbOwUCSCrstsiAQIDAZwERdEUSeFcWssIKHIlEeKt5MqqI+gb+md9738UfSkKdObsUiQlipSDSLC4nJ05Mzu33XUv9r1UdTp/XdrYfOfd996/fGXrgw8/+viTq9euP0+jceLIZ07kR8nLnp1K3wvlM+UpX76ME2kHPV++6I2OeP7FmUxSLwqP1XksTwN7EHp9z7EVkV5d2/ij25MDL5wqb/Q69hw1TuRsyzKfm9ZPkSvT+Xs3pNeTVNlKnlq3uk4U9qNx6BL+bctW1q1Oq3Pbmu4WJ3Zn96rEW66dDqXLKOMw6qUyOZNuCe8k8QZDZR1GfauId8r4lSIrNLEKRS5RgQwVwfakH02qYedsNVDRWDlRIMtAlfYA0rBngAXPPnQHRc/GthouulQSS8n6e2vxZyYuc69w9Rp66iVXayx4vpK3K0O3lHuvru502h18rOXBvhnsCPN5El278rPoCldEwhFjEQgpQqFo7AtbpPQ9EfuiI2KinYop0RIaeZiXYia2SHZMXJI4bKKO6HdAbyeGGtI7Y6aQdkiLT38JSVripuFxadwHVT9Zv1XgXaVjCmy28ZyePYMZEFWJIVGb5DLO9eV69A0aVq1oDXexWo9WEoPCfnBKa+7T06d3RSvk33PilDRySSqhkUM0n6iawjoSemrPs2+GiIQNPkkjtrrO7jqreT6i74iwbBqnsJRttcQjE5cQmiVsZR4fMV2N+DutUNtXh9Wfr8FD3PUqmPeYKCPxmkZl5DqdxexazaUM8qxCF/MrcITEnRIlQuZ75D+POMq56hAm67ARyQHWEiOD2gb5B8xn2RxTPPdgT4qoWcg3D3piU1m53ZmdPj17wE5IfkpzQ+hhX7TIHgnsBNmbebEF7oTeJnhzYKOzQG+jMjmeLVoTZ1eLcCOa8eZY2hPan5lF2rop0Szz3RM/Qr+kuLfgFc7rFslGsFBncQpdoYn9IfUV9ktAv0xlz2aUFnQEsGOIbBkRLdenkVjf17SKFPXjG9QpPX14LQZKC9rZPxOMA6yJIzqG7gmNXehhaYt6XVvcMXbMSlrZNx6607JWC/nHtas746JWlgtRqZbJykNo6ohvzYo5Amx30WuO6Ya6J6dY3WzBqh7sz7sG52jZPpbK41OOlYtsHEKqjMterF5n1VoOUCctxJJ9PyCOQ0j1DT1FbcZG01ah/o5MlUSIh4NnYHJC1ztrmyzMTIkeGE/5xKfrnPcknvmNMG3kVBfeK87XIT408tluwB1/Cmq93HNUVlk2odF0PlMv72Ek4R894ohrvBi+jrGWL+d/VmFmXey3gxcuIXLGKoO0LrqL3Khe+bHYoV6wQ1lV9idHmikJTicxjXfnnLuEy7wjaAmJwhU+nc/PGmPwgCgz47E+5rI6zG3w0PlccHZN5mkOXVFqgYOrul4r52VV3mh6vaw7z/eyrKbXyzLlDLnpmX6QAutlg5xCHFgiKGTpcYNUhD1B07XNv6xhX4i9dmzyIjEWvmiQDNBj9KoiU5WPG+1T6IKJ8QnL/PoGHjxD/5vhBHRRP+ay6kLezOXO38inufzkgp7NJYML+jeXfN0gKbE3eHMayzxtrCnmetLIpfOpaU+QtJtk5xddbS389siLvGe7pjNZZr/hrNC74yF6QGut/ry873Al7zfW8Rjdt4c+lECzW1M7z94aWh7BsdnLs908O4WnhT7vQWN+Qp/iNDUxO0J2rtqDJ7Uvvqee8pjOKjw+Et/QOWmffh+Vdo91Ufm+MjZ9p4h7QMh3qO8/wP5wcdzY3Cr2SnePso6HpCH7m5lbWs7tm7P0KqSL6K1G1tFzl870i5r0qW1IXPomc96AmOdDNV59fKuR9X8cIpyk8/vM20Au3tXWw8tzZvXNuum+v/oWGtOzT78+OmNqeB8g63x0SEm1qm/K+hRytHTn5/w9oKrg3J3euL6gr4y/eIsc4BY4Jo6cR98GFKQ1rYi46j4aQ0aZU1eKu1t+Q22Xbpwab9G2tJBXgcE7NB1Kmq679erqzv7if7GWB88P2vvftTtPD3bu3zX/4bosPhc3xC3y0x1xnyrzCfnV2fhn84vNm5tfbf+5/ff2v9v/adaNS0bmU1H6fPbe/3rTp4E=</latexit>

W<latexit sha1_base64="oLNh5jgGL8lSG3RxxH8XLmVD2Fg=">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</latexit>

U

T := ↵w W + ↵u U

Y :=

wW +

uU +

Proof coming
after next slideTT<latexit sha1_base64="ECX4aNrFkygg1eWzyV7O7iEbFzI=">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</latexit>

E[Y (1)Y

Y (0)] = EW,U [E[Y | T = 1, W, U ]<latexit sha1_base64="zWr5HNn3VSEmW3hoOPrylTWI2uA=">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</latexit>

E[Y | T = 0, W, U ]] =?

<latexit sha1_base64="bCgkyUx3yWUkqyiD0hPOxiEVoeE=">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</latexit><latexit sha1_base64="bCgkyUx3yWUkqyiD0hPOxiEVoeE=">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</latexit>

EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] = +u↵u<latexit sha1_base64="TLdb9LB2IawHGV9IHLw7pLDpis0=">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</latexit><latexit sha1_base64="TLdb9LB2IawHGV9IHLw7pLDpis0=">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</latexit>

Bias of EW [E[Y | T = 1, W ]Brady NealE[Y | T = 0, W ]] = +Sensitivity Analysis: Linear Single Confounder

u

↵u

=

u

↵u
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Contour Plots for Sensitivity to Confounding

Brady Neal

Sensitivity Analysis: Linear Single Confounder

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Contour Plots for Sensitivity to Confounding<latexit sha1_base64="LcpGuqRnUiblZeChoOywKImqzxE=">AAAUl3icrVhZb9tGEF6nV+JeTuqXoi9snAJ9kFXJTZG+GE1rOwiKBkgK56ppGBS5kgjzCrmy4xD6oUX/TGe+XYqkDlIOIkHianbmm9m5dleDJPAz1ev9u3Hjo48/+fSzm7c2P//iy6++3rp950UWT1JXPnfjIE5fDZxMBn4knytfBfJVkkonHATy5eD8gOdfXsg08+PoWF0l8jR0RpE/9F1HEels6z9bybcq/8N3MiseWlPLPjrLbTeOhvEk8kiQKIEcqhOaOLHjiXLjUFp26HuWrUiRCmWkrH2r37GqUqfWbotAb17ATv3RWJ3SlD1MHTe3B1I5Z/YkigeZTC+kV+G+cIJpbjtBMm7iONva6XV7eFmLg74Z7AjzehrfvvW3sIUnYuGKiQiFFJFQNA6EIzJ6n4i+6ImEaKciJ1pKIx/zUkzFJslOiEsSh0PUc/oe0a8TQ43oN2NmkHZJS0CflCQt8YPh8Wg8BFU/Wb9V4V2lIwc223hFz4HBDImqxJiobXIF5/pyA3qHLatWtIZfsVqfVpKAwn5wa2se0jOg34pWyN9XxClp5JFUSiOXaAFRNYV1pPTUnmffjBEJB3ySRmx1k91NVvN8TO9zwnJonMFSttUSj0xcImiWsJV5AsR0NeJbWqG2rwlrOFuDj7jrVTDvMVHOxTsa1ZGbdFazazWXMsjTJbqYX4EjIu6MKDEy3yf/+cRRz1WXMFmHg0iOsJYEGdQ1yH9ivsjmhOK5C3syRM1CvvnQk5jKKu0u7AzoOQB2SvI5zY2hh33RIXsksFNkb+HFDrhT+nWJXy5sdOfoXVQmx7NDa+Ls6hBuTDP+DEt7QvuzsEhblxPNMu9d8Rf0S4p7B17hvO6QbAwLdRZn0BWZ2O9TX2G/hPTNVPZsQelARwg7xsiWc6KV+jQS6/uJVpGhfgKDmtMzgNcSoHSgnf1ziXGINXFEJ9B9SWMPeljaol7XFQ+MHdOaVvaNj+60qNVC/nHt6s44r5XlIlSqZbJyH5p64r5ZMUeA7a56zTXdUPfkDKubzlk1gP1l1+AcrdvHUmV86rHykI1jSNVx2YvL17lsLXuokw5iyb4fEcc+pIaGnqE2E6Nps1J/B6ZKYsTDxTM0OaHrnbVdzs3kRA+NpwLi03XOexLPvCFMBzllw3vV+SbEIyNf7Abc8XNQm+VeoLLqsimN8tlMs7yPkYR/9IgjrvES+DrBWr6ffazKzLrYHwYvWkDkjFUGaV10D7mxfOXHYod6wQ5lVd2fHGmmpDidJDS+N+O8R7jMew4tEVG4wvPZ/LQ1BodEmRqPDTFX1GFpg4/O54HTNpmnOXRFqTkOrupmrZyXy/JG05tlvVm+12U1vVmWKRfITd/0gwxYr1rkFOLAEmElS49bpGLsCZqubX69hn0R9tqJyYvUWPiyRTJEj9Grik1VPmm1T6ELpsYnLPPPe3jwAv1vihPQdf1YyqprebOUu3ovn5byl9f0bCkZXtO/peS7FkmJvcGf0VjmWWtNMdfTVi6dT217gqTdpDi/6Grr4HtAXuQ92zOdyTL7DWeF3h330QM6a/XnxX2HK7nfWscTdN8B+lAKzV5D7Tz/YGhlBCdmLy928+IUnlX6vA+N5Qk9x2nq0uwIxblqF57UvnhMPeUJnVV4fCB+pnNSn74f1XaPdVH5vjIxfaeKu0fID6jvH2J/uD5uYm4Vu7W7R13HEWkoPlNzSyu5A3OWXoV0Hb3LkXX0vIUz/bwmfWobE5e+yVy1IJb5sByvOb7LkfU/DjFO0uV95kMgV+9q6+GVObP6Zt123199C03oOaTvAJ0xM7yHyLoAHVJSreqbsj6FHCzc+Tl/96gqOHfzu3fm9NXx52+RI9wCJ8RR8ujbgIK0plURV91HE8goc+rKcHcrb6jd2o1T483bllXyKjR4+6ZDSdN1N8+2dvrz/2ItDl7sdfu/dHvP7u88vG/+4bopvhN3xY/kpwfiIVXmU/Kru/H7xmgj2Xiz/e32b9uPth9r1hsbRuYbUXttP/sfa3cgnQ==</latexit>

Bias of EW [E[Y | T = 1, W ]Brady NealE[Y | T = 0, W ]] =u↵uSensitivity Analysis: Linear Single Confounder

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Contour Plots for Sensitivity to Confounding<latexit sha1_base64="LcpGuqRnUiblZeChoOywKImqzxE=">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</latexit>

Bias of EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] =u↵u<latexit sha1_base64="jc7ZpVoZh/c/adpaeMiCombd7k0=">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</latexit>u<latexit sha1_base64="0L4n8oCOuZcyRjnIgv9VZIKeFb8=">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</latexit>

Brady Neal

1↵u

Sensitivity Analysis: Linear Single Confounder

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Contour Plots for Sensitivity to Confounding<latexit sha1_base64="LcpGuqRnUiblZeChoOywKImqzxE=">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</latexit>

Bias of EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] =u↵u<latexit sha1_base64="P19Fby+XaPzio8piT7yEV2vZ8bA=">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</latexit>u↵u<latexit sha1_base64="jc7ZpVoZh/c/adpaeMiCombd7k0=">AAATwXicrVjbbttGEN2kt8S9OdFjX9QoBfogK5Lrwn0xEMBOUBQN4BS+tXFgSBRlEeKtJGXVEfQRfW2/rH/TM2eXIqkLKQeRIHE1O3Nmdm67q17oOnHSbv937/5HH3/y6WcPHm59/sWXX329/ejxWRyMI8s+tQI3iC563dh2Hd8+TZzEtS/CyO56Pdc+740OZf78xo5iJ/BPktvQfut1r31n4FjdBKTzy56ddK/GV9uNdqvNV3150DGDhjKv4+DRw9/UpeqrQFlqrDxlK18lGLuqq2K836iOaqsQtLdqClqEkcN5W83UFmTH4LLB0QV1hO9r/HpjqD5+C2ZMaQtaXHwiSNbVd4anj/GAVP0U/fUc7zodU2KLjbd49gymB2qihqBWyaWcm8v18PYqVp1gDT9xtQ5WEpIifrAKax7g6eJ3ghXK9y04bYz6kIowskBzQdUU0RHhqT0vvhkyEl3y2RiJ1WV2l1kt8wHeI2B1MY5pqdhaVy9NXHxqtmmr8LiM6XrEv7BCbV8Z1mC+Bodx16sQ3hNQRuodRkXkMp357FrPlRjk2Qpdwp+Qwwd3DErAzHfgPwccxVy1gCk6uozkNdcSMoNaBvkXzqfZHCKeO7QnZtTqzDeHekJTWZndqZ0unj1iR5CfYm5IPeKLJuyxiR0xe1MvNskd4deEvyzaaC3QW6xMiWcTa5LsagI3wIwzx9Ke0P5MLdLWTUGrm/eO+pX6bcS9Sa9IXjchG9BCncUxdfkm9gfoK+IXD99CFc+mlCZ1eLRjyGwZgZbp00ii7xlWEbN+XIM6xdOl10KiNKld/DPh2OOaJKJj6p5g3Kceka6j17XUvrFjVtAqvnHYnZa11pl/Uru6My5qFTmflVo3WXlATW21Z1YsERC7816zTDfUPTnm6mYLVvVof9Y1JEeL9olUFp9irPrMxiGlirjixdXrXLWWXdZJk7EU31+D44BSA0OPWZuh0bSVq79DUyUB42Hx6Zmc0PUu2iYLM1PQPeMpF3y6zmVPkpk/gdllTl3Se/n5MsQXRj7dDaTjT0ktlztjZRVlI4ym85lyeYcjm/7RI4m4xgvp65Br+Xb+qedmNsX+MHj+EqJkbGKQNkXvMzdWr/xENdALGsiqoj8l0kKJeDoJMX4653wKXOEdUYsPilT4dD4/q4zBESgz47EB59I6zGxw2Pn65Lw0mac5dEUlCxxS1eVaJS9X5Y2ml8v25/lelNX0clmh3DA3HdMPYmJdVMgljINIeLksPamQCrgnaLq2+fcN7PO5145NXkTGwvMKSY89Rq8qMFX5qtK+hF0wMj4RmT/ew4M37H8znoDu6sdMNrmTNzO52/fyaSY/uaNnM0nvjv7NJN9VSNrcG5w5TWReV9aUcB1Xcul8qtoTbOwm6flFV1uT3z14UfbsvulMdbPfSFbo3fGAPaC5UX9e3nekkjuVdTxm9+2xD0XU3C+pndMPhpZFcGz28nQ3T0/hca7PO9SYndCnPE1NzI6Qnqt26Enti5/RU17hrCLjQ/UDzkkdfL8s7B6bosp9ZWz6Th53F8j76PtH3B/ujhuaW8VO4e5R1PECGtLPzNzSMm7XnKXXId1F72pkHb3+0pl+UZM+tQ3BpW8ytxWIWT6sxiuP72pk/Y9DwJN0dp/5EMj5u9pmeFnOrL9ZV933199CQzwH+HbZGWPDe8Ssc9khbdSqvinrU8jh0p1f8ncXVSG5O33yeEFfEX/xFnnNW+AYHBmPvg0klNa0POK6+2hImcScumLe3bIbaqtw49R4i7bFubzyDN6B6VC26bpbV9uNzuK/WMuDs91W58dW+/Ve4/me+YfrgfpGPVHfw0/76jkq8xh+lUj8rf5R/9YOa04trEWa9f49I1NThVdt+j8CQ9El</latexit>u<latexit sha1_base64="0L4n8oCOuZcyRjnIgv9VZIKeFb8=">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</latexit>

Brady Neal

1↵u

Sensitivity Analysis: Linear Single Confounder

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Contour Plots for Sensitivity to Confounding<latexit sha1_base64="LcpGuqRnUiblZeChoOywKImqzxE=">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</latexit>

Bias of EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] =u↵u<latexit sha1_base64="P19Fby+XaPzio8piT7yEV2vZ8bA=">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</latexit>u↵u<latexit sha1_base64="jc7ZpVoZh/c/adpaeMiCombd7k0=">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</latexit>u<latexit sha1_base64="0L4n8oCOuZcyRjnIgv9VZIKeFb8=">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</latexit>

Brady Neal

1↵u

Sensitivity Analysis: Linear Single Confounder

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Contour Plots for Sensitivity to Confounding<latexit sha1_base64="LcpGuqRnUiblZeChoOywKImqzxE=">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</latexit>

Bias of EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] =u↵u<latexit sha1_base64="P19Fby+XaPzio8piT7yEV2vZ8bA=">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</latexit>u↵u<latexit sha1_base64="jc7ZpVoZh/c/adpaeMiCombd7k0=">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</latexit>u<latexit sha1_base64="0L4n8oCOuZcyRjnIgv9VZIKeFb8=">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</latexit>

Brady Neal

1↵u

Sensitivity Analysis: Linear Single Confounder

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Contour Plots for Sensitivity to Confounding<latexit sha1_base64="LcpGuqRnUiblZeChoOywKImqzxE=">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</latexit>

Bias of EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] =u↵u<latexit 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sha1_base64="jc7ZpVoZh/c/adpaeMiCombd7k0=">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</latexit>u<latexit sha1_base64="0L4n8oCOuZcyRjnIgv9VZIKeFb8=">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</latexit>

Brady Neal

1↵u

Sensitivity Analysis: Linear Single Confounder

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Contour Plots for Sensitivity to Confounding<latexit sha1_base64="EVQl4N7kYJF3METdJqxOLrx8KMA=">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</latexit><latexit sha1_base64="LcpGuqRnUiblZeChoOywKImqzxE=">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</latexit>

Bias of EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] =u

= EW [E[Y | T = 1, W ]E[Y | T = 0, W ]]

↵u

u

↵u

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Brady Neal1↵u<latexit sha1_base64="0L4n8oCOuZcyRjnIgv9VZIKeFb8=">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</latexit>

Sensitivity Analysis: Linear Single Confounder1↵u37 / 48

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Bias of EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] =u

= EW [E[Y | T = 1, W ]E[Y | T = 0, W ]]

↵u

u

↵u

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Brady Neal

1
↵u

= 25
<latexit sha1_base64="0L4n8oCOuZcyRjnIgv9VZIKeFb8=">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</latexit><latexit sha1_base64="npInmf1dGy3KD2NGzBZ/YIbVCPA=">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</latexit>

EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] = 25

Sensitivity Analysis: Linear Single Confounder1↵u37 / 48

Contour Plots for Sensitivity to Confounding<latexit sha1_base64="MeeE36HURkmUSR1Z1K/ClSRPLTE=">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</latexit><latexit sha1_base64="LcpGuqRnUiblZeChoOywKImqzxE=">AAAUl3icrVhZb9tGEF6nV+JeTuqXoi9snAJ9kFXJTZG+GE1rOwiKBkgK56ppGBS5kgjzCrmy4xD6oUX/TGe+XYqkDlIOIkHianbmm9m5dleDJPAz1ev9u3Hjo48/+fSzm7c2P//iy6++3rp950UWT1JXPnfjIE5fDZxMBn4knytfBfJVkkonHATy5eD8gOdfXsg08+PoWF0l8jR0RpE/9F1HEels6z9bybcq/8N3MiseWlPLPjrLbTeOhvEk8kiQKIEcqhOaOLHjiXLjUFp26HuWrUiRCmWkrH2r37GqUqfWbotAb17ATv3RWJ3SlD1MHTe3B1I5Z/YkigeZTC+kV+G+cIJpbjtBMm7iONva6XV7eFmLg74Z7AjzehrfvvW3sIUnYuGKiQiFFJFQNA6EIzJ6n4i+6ImEaKciJ1pKIx/zUkzFJslOiEsSh0PUc/oe0a8TQ43oN2NmkHZJS0CflCQt8YPh8Wg8BFU/Wb9V4V2lIwc223hFz4HBDImqxJiobXIF5/pyA3qHLatWtIZfsVqfVpKAwn5wa2se0jOg34pWyN9XxClp5JFUSiOXaAFRNYV1pPTUnmffjBEJB3ySRmx1k91NVvN8TO9zwnJonMFSttUSj0xcImiWsJV5AsR0NeJbWqG2rwlrOFuDj7jrVTDvMVHOxTsa1ZGbdFazazWXMsjTJbqYX4EjIu6MKDEy3yf/+cRRz1WXMFmHg0iOsJYEGdQ1yH9ivsjmhOK5C3syRM1CvvnQk5jKKu0u7AzoOQB2SvI5zY2hh33RIXsksFNkb+HFDrhT+nWJXy5sdOfoXVQmx7NDa+Ls6hBuTDP+DEt7QvuzsEhblxPNMu9d8Rf0S4p7B17hvO6QbAwLdRZn0BWZ2O9TX2G/hPTNVPZsQelARwg7xsiWc6KV+jQS6/uJVpGhfgKDmtMzgNcSoHSgnf1ziXGINXFEJ9B9SWMPeljaol7XFQ+MHdOaVvaNj+60qNVC/nHt6s44r5XlIlSqZbJyH5p64r5ZMUeA7a56zTXdUPfkDKubzlk1gP1l1+AcrdvHUmV86rHykI1jSNVx2YvL17lsLXuokw5iyb4fEcc+pIaGnqE2E6Nps1J/B6ZKYsTDxTM0OaHrnbVdzs3kRA+NpwLi03XOexLPvCFMBzllw3vV+SbEIyNf7Abc8XNQm+VeoLLqsimN8tlMs7yPkYR/9IgjrvES+DrBWr6ffazKzLrYHwYvWkDkjFUGaV10D7mxfOXHYod6wQ5lVd2fHGmmpDidJDS+N+O8R7jMew4tEVG4wvPZ/LQ1BodEmRqPDTFX1GFpg4/O54HTNpmnOXRFqTkOrupmrZyXy/JG05tlvVm+12U1vVmWKRfITd/0gwxYr1rkFOLAEmElS49bpGLsCZqubX69hn0R9tqJyYvUWPiyRTJEj9Grik1VPmm1T6ELpsYnLPPPe3jwAv1vihPQdf1YyqprebOUu3ovn5byl9f0bCkZXtO/peS7FkmJvcGf0VjmWWtNMdfTVi6dT217gqTdpDi/6Grr4HtAXuQ92zOdyTL7DWeF3h330QM6a/XnxX2HK7nfWscTdN8B+lAKzV5D7Tz/YGhlBCdmLy928+IUnlX6vA+N5Qk9x2nq0uwIxblqF57UvnhMPeUJnVV4fCB+pnNSn74f1XaPdVH5vjIxfaeKu0fID6jvH2J/uD5uYm4Vu7W7R13HEWkoPlNzSyu5A3OWXoV0Hb3LkXX0vIUz/bwmfWobE5e+yVy1IJb5sByvOb7LkfU/DjFO0uV95kMgV+9q6+GVObP6Zt123199C03oOaTvAJ0xM7yHyLoAHVJSreqbsj6FHCzc+Tl/96gqOHfzu3fm9NXx52+RI9wCJ8RR8ujbgIK0plURV91HE8goc+rKcHcrb6jd2o1T483bllXyKjR4+6ZDSdN1N8+2dvrz/2ItDl7sdfu/dHvP7u88vG/+4bopvhN3xY/kpwfiIVXmU/Kru/H7xmgj2Xiz/e32b9uPth9r1hsbRuYbUXttP/sfa3cgnQ==</latexit>

Bias of EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] =

u

u

= 25

↵u

↵u

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Brady Neal

1
↵u

= 25
<latexit sha1_base64="0L4n8oCOuZcyRjnIgv9VZIKeFb8=">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</latexit><latexit sha1_base64="npInmf1dGy3KD2NGzBZ/YIbVCPA=">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</latexit>

EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] = 25

Sensitivity Analysis: Linear Single Confounder1↵u37 / 48

Contour Plots for Sensitivity to Confounding<latexit sha1_base64="MeeE36HURkmUSR1Z1K/ClSRPLTE=">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</latexit><latexit sha1_base64="LcpGuqRnUiblZeChoOywKImqzxE=">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</latexit>

Bias of EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] =

u

u

= 25

↵u

↵u

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sha1_base64="0L4n8oCOuZcyRjnIgv9VZIKeFb8=">AAATznicrVjbbttGEN2kt8S9OdZjX9QoBfogK5Lrwn0xEMBOUBQN4BS+tXFgUBRlESJFlqSsOoTQ135FX9tf6t/0zNmlKOpGKYgEiavZmTOzc9tdtUPPjZNm87979z/48KOPP3nwcOvTzz7/4svtRzvncTCMbOfMDrwgumxbseO5A+cscRPPuQwjx/LbnnPR7h/J/MWtE8VuMDhN7kLnjW/dDNyua1sJSNfbO1fdyLLT1ji9srywZ10Px9fbtWajyVd1ftAyg5oyr5Pg0cNf1JXqqEDZaqh85aiBSjD2lKVivF+rlmqqELQ3KgUtwsjlvKPGaguyQ3A54LBA7eP7Br9eG+oAvwUzprQNLR4+ESSr6hvD08G4S6p+iv7qFO8yHSmxxcY7PNsG0wc1UT1Qy+QyzvXl2nj7JatOsIYfuFoXKwlJET/YhTV38fTwO8EK5fsOnA5GHUhFGNmgeaBqiuiI8NSeF9/0GAmLfA5GYvUqu1dZLfMB3n1gWRjHtFRsraoXJi4DanZoq/B4jOlyxD+wQm3fKqzuZA0u465XIbynoPTVW4yKyKt0TmfXcq7EII8X6BL+hBwDcMegBMx8F/5zwVHMVRuYosNiJG+4lpAZ1DDIP3E+y+YQ8dylPTGjVmW+udQTmsrK7c7s9PBsEzuCfIq5HvWIL+qwxyF2xOzNvFgnd4RfI/6yaaM9Q2+wMiWedaxJsqsO3AAz7gRLe0L7M7NIW5eCVjXvXfUz9TuIe51ekbyuQzaghTqLY+oamNgfoq+IX3x8C1U8m1Hq1OHTjh6zpQ9ark8jib6nWEXM+vEMaoqnR6+FRKlTu/hnxLHPNUlEh9Q9wrhDPSJdRa9rqANjx7igVXzjsjvNa60y/6R2dWec1SpyA1Zq1WTlITU11b5ZsURA7J72mm26oe7JMVc3nrGqTfvzriE5WrRPpPL4FGPVYTb2KFXEFS8uXueiteyxTuqMpfj+BhyHlOoaeszaDI2mran6OzJVEjAeNp++yQld76JtNDOTgu4bT3ng03Uue5LM/A5Mizl1Re9Nz69CfG7ks91AOn5K6mq5c1ZWUTbCKJ3MrJZ3OXLoHz2SiGu8kL4OuZavJ5/q1My62O8HbzCHKBmbGKR10TvMjcUrP1U19IIasqroT4m0UCKeTkKMn0w4nwBXePvUMgBFKjydzI9LY3AMyth4rMu5rA5zG1x2vg45r0zmaQ5dUckMh1T1aq2Sl4vyRtNXy3Ym+V6U1fTVskK5ZW66ph/ExLoskUsYB5Hwp7L0tEQq4J6g6drmX9ewb8C9dmjyIjIWXpRI+uwxelWBqcqXpfYl7IKR8YnI/PYOHrxl/xvzBLSpH3PZZCNv5nJ37+TTXH60oWdzSX9D/+aSb0skHe4N7oQmMq9Ka0q4Tkq5dD6V7QkOdpPs/KKrrc7vNrwoe3bHdKaq2W8kK/TueMgeUF+rP8/vO1LJrdI6HrL7ttmHImrurKids/eGlkdwaPbybDfPTuHxVJ93qTE/oac8TY3MjpCdq3bpSe2LH9FTXuKsIuMj9R3OSS18vyjsHuuiyn1laPrONO4ekA/Q94+5P2yOG5pbxW7h7lHU8Rwass/Y3NJybs+cpZchbaJ3MbKOXmfuTD+rSZ/aeuDSN5m7EsQ8HxbjrY7vYmT9j0PAk3R+n3kfyNN3tfXw8pxZfrMuu+8vv4WGeHbx7bEzxob3mFnnsUM6qFV9U9ankKO5O7/k7x6qQnI3fbwzo6+IP3uLvOEtcAiOnEffBhJKa9o04rL7aEiZxJy6Yt7d8htqo3Dj1HiztsVTeeUbvEPToRzTdbeut2ut2X+x5gfne43W943mq/3as33zD9cD9ZV6rL6Fnw7UM1TmCfxqY3f+W/2j/q2cVG4r48qfmvX+PSNTUYVX5a//AXk/1kc=</latexit>

Brady Neal

1
↵u

= 25
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EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] = 25

Sensitivity Analysis: Linear Single Confounder1↵u37 / 48

Bias in Simple Linear Setting Proof: Outline

Brady Neal

Sensitivity Analysis: Linear Single Confounder

38 / 48

Bias in Simple Linear Setting Proof: Outline<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:

Brady Neal

T := ↵w W + ↵u U
Y :=

wW +

uU +

T

Sensitivity Analysis: Linear Single Confounder

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Bias in Simple Linear Setting Proof: Outline<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:

T := ↵w W + ↵u U
Y :=

wW +

uU +

Result:
The confounding bias of adjusting for just W (and not U) is<latexit sha1_base64="deDm4GbQFFe8Pui/gkmmuey3YMw=">AAAVbXicrVjpbttGEN64V+JeSVqjP1oUbJym/SGrkpsi/SMggOO0KBogKZyjNQODx8oizCs87DqE3qmv06foK3Tm26VI6iClIBIkrmZnvpmda3dlx76XZoPBv1e23nn3vfc/uHpt+8OPPv7k0+s3bj5Lozxx5FMn8qPkhW2l0vdC+TTzMl++iBNpBbYvn9tnBzz//FwmqReFR9llLF8G1mnojT3Hyoh0cuPKP+bhSWE6UTiO8tAlzqlxxzB9Oc6ODfPw2IzyzIkCaZiB5xpmRthZIMPMGBnDnlGXe2nsdQgM5gXMxDudZPQMozAPbJkYprlt0OuO+Sq3XAA2jSOEPIzsVCbn0m0Y/SYmr0LbeCmrgcoljgxznFhOYdoys06Wc59b/rQwLT+etHGcXN8d9Ad4GYuDoR7sCv16HN249ocwhSsi4YhcBEKKUGQ09oUlUnofi6EYiJhoL0VBtIRGHualmIptks2JSxKHRdQz+j6lX8eaGtJvxkwh7ZAWnz4JSRriW83j0ngMqnqyfqPGu0pHAWy28ZKetsYMiJqJCVG75ErO9eVsegcdq85oDT9jtR6tJAaF/eA01jymp0+/M1ohf18Sp6SRS1IJjRyi+URVFNaR0FN5nn0zQSQs8EkasdVtdrdZzfMRvc8Iy6JxCkvZVkM81HEJoVnCVubxEdPViH/TCpV9bVjj2Ro8xF2tgnmPiHImXtOoidyms55dq7kyjTxdoov5M3CExJ0SJULme+Q/jziaueoQJuuwEMlTrCVGBvU18m+YL7M5pnjuwZ4UUTOQbx70xLqyKrtLO3162sBOSL6guQn0sC96ZI8EdoLsLb3YA3dCvy7wy4GNzhy9j8rkePZoTZxdPcKNaMabYSlPKH+WFinrCqIZ+r0nfod+SXHvwSuc1z2SjWChyuIUukId+xH1FfZLQN9MZc+WlB50BLBjgmw5I1qlTyGxvh9oFSnqx9eoBT19eC0GSg/a2T8XGAdYE0c0h+4LGrvQw9IG9bq+uKftmDa0sm88dKdFrQbyj2tXdcZ5rSwXolINnZUjaBqIu3rFHAG2u+41R3dD1ZNTrG46Z5UN+6uuwTnatI+lqvg0Y+UiGyeQauKyF5evc9la9lEnPcSSfX9KHCNIjTU9RW3GWtN2rf4OdJVEiIeDZ6BzQtU7a7uYmymIHmhP+cSn6pz3JJ55RZgWcsqE9+rzbYiHWr7cDbjjF6C2yz1DZTVlExoVs5l2eQ8jCf+oEUdc4cXwdYy1fDP7GLWZdbHfDl64gMgZm2mkddFd5MbylR+JXeoFu5RVTX9ypJmS4HQS0/j2jPM24TLvGbSEROEKL2bz084YPCDKVHtsjLmyDisbPHQ+F5ymzjzFoSoqm+Pgqm7Xynm5LG8UvV3WneV7U1bR22WZco7c9HQ/SIH1okMuQxxYIqhl6VGHVIQ9QdGVzX+uYV+IvTbXeZFoC593SAboMWpVka7KR532ZeiCifYJy/z1Bh48R/+b4gS0qR8r2Wwjb1Zyl2/k00r+YkPPVpLBhv6tJF93SErsDd6MxjJPOmuKuR53cql86toTJO0m5flFVVsP3zZ5kfdsV3cmQ+83nBVqdxyhB/TW6s+L+w5X8rCzjnN0Xxt9KIFmt6V2nr41tCqCud7Ly928PIWntT7vQWN1Qi9wmrrQO0J5rtqDJ5UvfqWe8ojOKjw+ED/SOWlI3w8bu8e6qHxfyXXfqePuE/I96vsPsD9sjhvrW8Ve4+7R1HFIGsrPVN/SKm5fn6VXIW2idzmyip67cKaf16RObRPiUjeZyw7EKh+W47XHdzmy+schwkm6us+8DeT6XW09vCpnVt+su+77q2+hMT3H9O2jM6aa9wGyzkeHlFSr6qasTiEHC3d+zt99qgrO3eLWzTl9Tfz5W+QpboE5cVQ86jaQQVrR6oir7qMxZDJ96kpxd6tuqP3GjVPhzduW1vIq0Hgj3aGk7rrbJ9d3h/P/Yi0Onu33hz/1B0/u7t6/q//huiq+FLfE9+Sne+I+VeZj8quz9cXWaOvh1i+f/7ezs/PVzteKdeuKlvlMNF473/0P4cxtXw==</latexit>

EW [E[Y | T = 1, W ]<latexit sha1_base64="P19Fby+XaPzio8piT7yEV2vZ8bA=">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</latexit>u↵u. Formally,E[Y | T = 0, W ]]EW,U [E[Y | T = 1, W, U ]Brady NealT

E[Y | T = 0, W, U ]] =Sensitivity Analysis: Linear Single Confounderu↵u

38 / 48

Bias in Simple Linear Setting Proof: Outline<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:

T := ↵w W + ↵u U
Y :=

wW +

uU +

Result:
The confounding bias of adjusting for just W (and not U) is<latexit sha1_base64="deDm4GbQFFe8Pui/gkmmuey3YMw=">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</latexit>

EW [E[Y | T = 1, W ]T<latexit sha1_base64="P19Fby+XaPzio8piT7yEV2vZ8bA=">AAAUBHicrVhLb9tGEN6kr8R9OdGxFzVKgR5kVXJduBcDAewEQdEATuFXGwUGHyuLEF8lKasOoWt/TdFL0Wt/RS89tL+lM98uRVIPUgosQeJqduab2XntrszQdeKk2/3nzt133n3v/Q/u3d/68KOPP/l0+8HDszgYR5Y8tQI3iC5MI5au48vTxElceRFG0vBMV56bo0OeP7+WUewE/klyE8rXnnHlOwPHMhIiXW4/7w8iw0r7pkyMy/7YD8xYRtfStgJ/EIx9m0SvDXea9g03HFZxXG63up0uXs3FQU8PWkK/joMH938QfWGLQFhiLDwhhS8SGrvCEDG9X4me6IqQaK9FSrSIRg7mpZiKLZIdE5ckDoOoI/q+ol+vNNWn34wZQ9oiLS59IpJsii80j03jAajqyfqbBd5VOlJgs4039DQ1pkfURAyJWieXca4vZ9Lbq1l1Qmv4Fqt1aCUhKOwHq7TmAT1d+p3QCvn7hjgljWySimhkEc0lqqKwjoieyvPsmyEiYYBP0oitrrK7ymqeD+g9IiyDxjEsZVub4pmOiw/NErYyj4uYrkb8hVao7KvCGszW4CDuahXMe0KUkXhDozJylc5idq3mSjTydIku5k/A4RN3TJQAme+Q/xziKOeqRZisw0Akr7CWEBnU0cjfYT7L5pDiuQN7YkStiXxzoCfUlZXbndnp0tMEdkTyKc0NoYd90SZ7JLAjZG/mxTa4I/o1wS8LNlpz9A4qk+PZpjVxdrUJN6AZZ4alPKH8mVmkrEuJ1tTvHfE99EuKexte4bxuk2wAC1UWx9Dl69gfUF9hv3j0zVT2bEZpQ4cHO4bIlhHRcn0KifV9RauIUT+uRk3p6cJrIVDa0M7+mWDsYU0c0TF0T2hsQw9LN6nXdcS+tmNa0sq+cdCdFrU2kX9cu6ozzmtlOR+V2tRZeQBNXbGnV8wRYLuLXrN0N1Q9OcbqpnNWmbA/7xqco2X7WCqPTzlWNrJxCKkyLntx+TqXrWUXddJGLNn3V8RxAKmBpseozVBr2irU36GukgDxsPD0dE6oemdtk7mZlOie9pRLfKrOeU/imZ8J00BO9eG94nwV4lMtn+0G3PFTUKvlzlBZZdmIRulsplrewUjCP2rEEVd4IXwdYi2fzz7Nwsy62LeD5y8gcsYmGmlddBu5sXzlJ6JFvaBFWVX2J0eaKRFOJyGNH884HxMu846gxScKV3g6m5/WxuCIKFPtsQHmsjrMbXDQ+Wxw9nXmKQ5VUckcB1d1tVbOy2V5o+jVsvYs38uyil4ty5Rr5Kaj+0EMrIsauQRxYAmvkKUnNVIB9gRFVzb/uIZ9Pvbasc6LSFt4XiPpoceoVQW6Kl/U2pegC0baJyzz01t48Br9b4oT0KZ+zGWTjbyZy928lU9z+cmGns0lvQ39m0u+qZGU2BucGY1lXtbWFHMd13KpfKrbEyTtJtn5RVVbG98meZH3bFt3pqbebzgr1O54gB7QXqs/L+47XMm92joeo/ua6EMRNNsVtXN6a2h5BMd6L8928+wUHhf6vAON+Qk9xWlqoneE7Fy1A08qXzynnvKCzio8PhRf0zmpR9/PSrvHuqh8XxnrvlPE3SXkfer7R9gfNscN9a1ip3T3KOt4Shqyz1Tf0nJuV5+lVyFtonc5soqevXCmn9ekTm1D4lI3mZsaxDwfluNVx3c5svrHIcBJOr/P3AZy8a62Hl6eM6tv1nX3/dW30JCeA/p20RljzXuErHPRISXVqropq1PI4cKdn/N3l6qCczd99HBOXxl//hZ5hVvgmDhyHnUbSCCtaEXEVffREDKJPnXFuLvlN9RO6cap8OZtiwt55Wm8A92hpO66W5fbrd78v1iLg7PdTu+bTvflXuvJnv6H6574TDwSX5Kf9sUTqsxj8qslfhd/i3/Ff41fG781/mj8qVjv3tEyDVF6Nf76H0CR7xg=</latexit>u↵u. Formally,E[Y | T = 0, W ]]EW,U [E[Y | T = 1, W, U ]E[Y | T = 0, W, U ]] =u↵uProof Outline:
1. Get a closed-form expression for EW [E[Y | T = t, W ]] in terms of ↵w, ↵u,<latexit sha1_base64="hFu/fmcnwBX81CcQb7VGetfssaI=">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</latexit><latexit sha1_base64="w1ZhPiDncXR2L8yzYwlu/GNyM/Q=">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</latexit>

Brady Neal

Sensitivity Analysis: Linear Single Confounder<latexit sha1_base64="48CstAaUpFlmEG+nuOp3jWzujXU=">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</latexit><latexit sha1_base64="T+W/IhsnYnJtUIWYolY1YBFilwA=">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</latexit>

w, and<latexit sha1_base64="fSJ0y/3irtENRAe31P2x41TtLN8=">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</latexit>u.

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Bias in Simple Linear Setting Proof: Outline<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:

T := ↵w W + ↵u U
Y :=

wW +

uU +

Result:
The confounding bias of adjusting for just W (and not U) is<latexit sha1_base64="deDm4GbQFFe8Pui/gkmmuey3YMw=">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</latexit>

EW [E[Y | T = 1, W ]T<latexit sha1_base64="P19Fby+XaPzio8piT7yEV2vZ8bA=">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</latexit>u↵u. Formally,E[Y | T = 0, W ]]EW,U [E[Y | T = 1, W, U ]E[Y | T = 0, W, U ]] =u↵uProof Outline:
1. Get a closed-form expression for EW [E[Y | T = t, W ]] in terms of ↵w, ↵u,
2. Use step 1 to get a closed-form expression for the difference<latexit sha1_base64="hFu/fmcnwBX81CcQb7VGetfssaI=">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</latexit><latexit sha1_base64="w1ZhPiDncXR2L8yzYwlu/GNyM/Q=">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</latexit>

EW [E[Y | T = 1, W ]<latexit sha1_base64="mza1ctzR+qochFal+JYosUMtVg8=">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</latexit>

Brady Neal<latexit sha1_base64="48CstAaUpFlmEG+nuOp3jWzujXU=">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</latexit><latexit sha1_base64="T+W/IhsnYnJtUIWYolY1YBFilwA=">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</latexit>

w, and<latexit sha1_base64="fSJ0y/3irtENRAe31P2x41TtLN8=">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</latexit>u.

E[Y | T = 0, W ]]Sensitivity Analysis: Linear Single Confounder

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Bias in Simple Linear Setting Proof: Outline<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:

T := ↵w W + ↵u U
Y :=

wW +

uU +

Result:
The confounding bias of adjusting for just W (and not U) is<latexit sha1_base64="deDm4GbQFFe8Pui/gkmmuey3YMw=">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</latexit>

EW [E[Y | T = 1, W ]T<latexit sha1_base64="P19Fby+XaPzio8piT7yEV2vZ8bA=">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</latexit>u↵u. Formally,E[Y | T = 0, W ]]EW,U [E[Y | T = 1, W, U ]E[Y | T = 0, W, U ]] =u↵uProof Outline:
1. Get a closed-form expression for EW [E[Y | T = t, W ]] in terms of ↵w, ↵u,
2. Use step 1 to get a closed-form expression for the difference<latexit sha1_base64="hFu/fmcnwBX81CcQb7VGetfssaI=">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</latexit><latexit sha1_base64="w1ZhPiDncXR2L8yzYwlu/GNyM/Q=">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</latexit>

EW [E[Y | T = 1, W ]<latexit sha1_base64="mza1ctzR+qochFal+JYosUMtVg8=">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</latexit>

3. Subtract off EW,U [E[Y | T = 1, W, U ]<latexit sha1_base64="Za6uReEA61igoPD5hC6PLEGyoqc=">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</latexit>

Brady Neal<latexit sha1_base64="48CstAaUpFlmEG+nuOp3jWzujXU=">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</latexit><latexit sha1_base64="T+W/IhsnYnJtUIWYolY1YBFilwA=">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</latexit>

w, and<latexit sha1_base64="fSJ0y/3irtENRAe31P2x41TtLN8=">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</latexit>u.

E[Y | T = 0, W ]]E[Y | T = 0, W, U ]] =Sensitivity Analysis: Linear Single Confounder

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Bias in Simple Linear Setting Proof: Step 1<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:

T := ↵w W + ↵u U
Y :=

wW +

uU +

T

Get a closed-form expression for EW [E[Y | T = t, W ]] in terms of ↵w, ↵u,<latexit sha1_base64="hFu/fmcnwBX81CcQb7VGetfssaI=">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</latexit><latexit sha1_base64="w1ZhPiDncXR2L8yzYwlu/GNyM/Q=">AAAT0nicrVhbbxtFFJ6WWxtuaS3xwoupi8SDY+wQFF4iVUpaIUSlFOUGTRWt1+t45b2xu45JjB8Qr/wKXuEP8W/4zjezXq9v61SxZe/4zDnfOXNuM+N25LlJ2mz+d+/+O+++9/4HDx5ufPjRx598uvno8UkSDmLbObZDL4zP2lbieG7gHKdu6jlnUexYfttzTtv9fZk/vXLixA2Do/Q6ct741mXgdl3bSkG62Pzs3PKinnVxbodBNxwEHfBeWd7FZq3ZaPJVnR+0zKCmzOswfPTwJ3WuOipUthooXzkqUCnGnrJUgvdr1VJNFYH2Ro1AizFyOe+osdqA7ABcDjgsUPv4vsSv14Ya4LdgJpS2ocXDJ4ZkVX1peDoYd0nVT9FfneJdpmNEbLHxGs+2wfRBTVUP1DK5jHN9uTbefsmqU6zhO67WxUoiUsQPdmHNXTw9/E6xQvm+BqeDUQdSMUY2aB6omiI6Yjy158U3PUbCIp+DkVi9yu5VVst8iHcfWBbGCS0VW6vqhYlLQM0ObRUejzFdjvgbVqjtW4XVnazBZdz1KoT3CJS+usGoiLxK53R2LedKDfJ4gS7hT8kRgDsBJWTmu/CfC45irtrAFB0WI3nJtUTMoIZB/oHzWTZHiOcW7UkYtSrzzaWeyFRWbndmp4dnm9gx5EeY61GP+KIOexxix8zezIt1csf4NeQvmzbaM/QGK1PiWceaJLvqwA0x406wtCe0PzOLtHUj0KrmvaV+pH4Hca/TK5LXdciGtFBncUJdgYn9HvqK+MXHt1DFsxmlTh0+7egxW/qg5fo0kuj7GqtIWD+eQR3h6dFrEVHq1C7+GXLsc00S0QF1DzHuUI9IV9HrGmrX2DEuaBXfuOxO81qrzD+pXd0ZZ7WKXMBKrZqs3KOmptoxK5YIiN3TXrNNN9Q9OeHqxjNWtWl/3jUkR4v2iVQen2KsOszGHqWKuOLFxetctJZt1kmdsRTfX4Jjj1JdQ09Ym5HRtDFVf/umSkLGw+bTNzmh6120DWdmRqD7xlMe+HSdy54kM78C02JOndN70/OrEJ8b+Ww3kI4/InW13AkrqygbYzSazKyWdzly6B89kohrvIi+jriWLyaf6tTMuth3gxfMIUrGpgZpXfQOc2Pxyo9UDb2ghqwq+lMiLZSYp5MI46cTzqfAFd4+tQSgSIWPJvPj0hgcgDI2HutyLqvD3AaXna9DznOTeZpDV1Q6wyFVvVqr5OWivNH01bKdSb4XZTV9taxQrpibrukHCbHOSuRSxkEk/KksPSqRCrknaLq2+ec17Au41w5MXsTGwtMSSZ89Rq8qNFX5stS+lF0wNj4RmV/ewoNX7H9jnoBu68dcNr2VN3O567fyaS4/vKVnc0n/lv7NJW9KJB3uDe6EJjKvSmtKuA5LuXQ+le0JDnaT7Pyiq63O7za8KHt2x3SmqtlvJCv07rjHHlBfqz/P7ztSya3SOh6w+7bZh2Jq7qyoneM7Q8sjODB7ebabZ6fwZKrPu9SYn9BHPE0NzY6Qnau26Enti+/RU17irCLjffUNzkktfL8o7B7rosp9ZWD6zjTuNpB30fcPuD/cHjcyt4qtwt2jqOM5NGSfsbml5dyeOUsvQ7qN3sXIOnqduTP9rCZ9auuBS99krksQ83xYjLc6vouR9T8OIU/S+X3mLpCn72rr4eU5s/xmXXbfX34LjfDs4ttjZ0wM7wGzzmOHdFCr+qasTyH7c3d+yd9tVIXk7ujJ4xl9RfzZW+Qlb4EDcOQ8+jaQUlrTphGX3UcjyqTm1JXw7pbfUBuFG6fGm7Utmcor3+DtmQ7lmK67cbFZa83+izU/ONlutL5tNF/t1J7tmH+4HqjP1RP1Ffy0q56hMg/hV1v9rv5W/6h/K0eVm8oflT816/17RqaiCq/KX/8DoWjYJg==</latexit>

Brady Neal

Sensitivity Analysis: Linear Single Confounder<latexit sha1_base64="T+W/IhsnYnJtUIWYolY1YBFilwA=">AAAT0XicrVhbbxtFFJ6WWxtuaS2eeDF1kXhwjB2CwkukSkkrhKiUQm7QVJG9u45X3hu76xjXsoR45VfwCr+If8N3vpn1en1bp6ote8dnzvnOmXObGXciz03SZvO/O3ffefe99z+4d3/rw48+/uTT7QcPz5JwEFvOqRV6YXzRaSeO5wbOaeqmnnMRxU7b73jOead/KPPnN06cuGFwko4i55Xfvg7crmu1U5Cutj+77Dhp++rSCoNuOAhssN60vavtWrPR5Ku6OGiZQU2Z13H44P5P6lLZKlSWGihfOSpQKcaeaqsE75eqpZoqAu2VGoMWY+Ry3lETtQXZAbgccLRB7eP7Gr9eGmqA34KZUNqCFg+fGJJV9aXhsTHukqqfor86w7tKx5jYYuMIz47B9EFNVQ/UMrmMc3O5Dt5+yapTrOE7rtbFSiJSxA9WYc1dPD38TrFC+R6B08HIhlSMkQWaB6qmiI4YT+158U2PkWiTz8FIrF5n9zqrZT7Euw+sNsYJLRVbq+qZiUtAzQ5tFR6PMV2N+DtWqO1bh9WdrsFl3PUqhPcElL56jVEReZ3O2exazZUa5MkSXcKfkiMAdwJKyMx34T8XHMVctYApOtqM5DXXEjGDGgb5B85n2Rwhnju0J2HUqsw3l3oiU1m53ZmdHp4dYseQH2OuRz3iizrscYgdM3szL9bJHePXkL8s2mjN0RusTIlnHWuS7KoDN8SMO8XSntD+zCzS1o1Bq5r3jvqR+h3EvU6vSF7XIRvSQp3FCXUFJvYH6CviFx/fQhXPZpQ6dfi0o8ds6YOW69NIou9rrCJh/XgGdYynR69FRKlTu/hnyLHPNUlEB9Q9xNimHpGuotc11L6xY1LQKr5x2Z0WtVaZf1K7ujPOaxW5gJVaNVl5QE1NtWdWLBEQu2e9ZpluqHtywtVN5qzq0P68a0iOFu0TqTw+xVjZzMYepYq44sXl61y2ll3WSZ2xFN9fg+OAUl1DT1ibkdG0NVN/h6ZKQsbD4tM3OaHrXbQN52bGoPvGUx74dJ3LniQzvwGzzZy6pPdm59chPjXy2W4gHX9M6nq5M1ZWUTbGaDydWS/vcuTQP3okEdd4EX0dcS1fTD/VmZlNsd8OXrCAKBmbGqRN0W3mxvKVn6gaekENWVX0p0RaKDFPJxHGj6ecj4ErvH1qCUCRCh9P5yelMTgCZWI81uVcVoe5DS47n03OS5N5mkNXVDrHIVW9Xqvk5bK80fT1svY034uymr5eVig3zE3X9IOEWBclcinjIBL+TJaelEiF3BM0Xdv8ywb2BdxrByYvYmPheYmkzx6jVxWaqnxeal/KLhgbn4jMr2/gwRv2vwlPQLf1Yy6b3sqbudzojXyayw9v6dlc0r+lf3PJ1yWSDvcGd0oTmRelNSVcx6VcOp/K9gQHu0l2ftHVVud3B16UPds2nalq9hvJCr07HrAH1Dfqz4v7jlRyq7SOB+y+HfahmJrtNbVz+tbQ8ggOzF6e7ebZKTyZ6fMuNeYn9DFPU0OzI2Tnqh16Uvvie/SU5ziryPhQfYNzUgvfzwq7x6aocl8ZmL4zi7sL5H30/SPuD7fHjcytYqdw9yjqeAoN2Wdibmk5t2fO0quQbqN3ObKOnr1wpp/XpE9tPXDpm8yoBDHPh+V46+O7HFn/4xDyJJ3fZ94G8uxdbTO8PGdW36zL7vurb6ERnl18e+yMieE9YtZ57JAOalXflPUp5HDhzi/5u4uqkNwdP3o4p6+IP3+LvOYtcACOnEffBlJKa9os4qr7aESZ1Jy6Et7d8htqo3Dj1HjztiUzeeUbvAPToRzTdbeutmut+X+xFgdnu43Wt43mi73akz3zD9c99bl6pL6Cn/bVE1TmMfxqQdvf6h/1b+XnyqjyR+VPzXr3jpGpqMKr8tf/X0TXsg==</latexit><latexit sha1_base64="48CstAaUpFlmEG+nuOp3jWzujXU=">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</latexit>

w

, and

<latexit sha1_base64="fSJ0y/3irtENRAe31P2x41TtLN8=">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</latexit>u.

39 / 48

Bias in Simple Linear Setting Proof: Step 1<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:
<latexit sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">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</latexit>

EW [E[Y | T = t, W ]] = EW [E[ w W +T := ↵w W + ↵u U
Y :=

uU +

wW +

uU +

T

T | T = t, W ]]= EW [ w W + u E[U | T = t, W ] + t]✓◆t ↵w W= EW w W + u
+ t
↵u

u
u ↵w
= EW w W +
t
W+ t
↵u
↵u
u
u ↵w
= w E[W ] +t
E[W ] + t
↵
↵u
✓
◆ u✓
◆
↵
u
u w
=
+
t+
E[W ]
w
↵u
↵u

Brady Neal

Sensitivity Analysis: Linear Single Confounder

39 / 48

Bias in Simple Linear Setting Proof: Step 1<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:
<latexit sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">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</latexit>

EW [E[Y | T = t, W ]] = EW [E[ w W +T := ↵w W + ↵u U
Y :=

uU +

wW +

uU +

T

T | T = t, W ]]= EW [ w W + u E[U | T = t, W ] + t]✓◆t ↵w W= EW w W + u
+ t
↵u

u
u ↵w
= EW w W +
t
W+ t
↵u
↵u
u
u ↵w
= w E[W ] +t
E[W ] + t
↵
↵u
✓
◆ u✓
◆
↵
u
u w
=
+
t+
E[W ]
w
↵u
↵u

Brady Neal

Sensitivity Analysis: Linear Single Confounder

39 / 48

Bias in Simple Linear Setting Proof: Step 1<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:
<latexit sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">AAAZ+Xic1VhbbxtFFJ4WCG24teUp4mVpCiqqY5xQVB6IVClNhRCVWpReIBtF691xvMreurtOSFf+MYgXxCu/gld44t9wzjez3kvWu3YVkLBl7/jMOd+5zDlnZjyMPDdJB4O/L11+4823Vt6+cnX1nXffe/+Da9dvPEvCSWzLp3bohfGLoZVIzw3k09RNPfkiiqXlDz35fHi8w/PPT2ScuGGwl55F8sC3jgJ35NpWSqTD6ytfm7uHmWmHwSicBA5xTg3Tk6N03zB3981wktqhLw3Tdx3DTAk59WWQGtulHyeW1zPKCAeGGbtH4/TA+JT42uCHMrUOy7OEVYEy7hiaaRKEw0TGJ9KpszfOsKAjvdQqW/2aXpjmqkGvVmcuxBOKSLMzSxre4DvwC3/+fV8Y67ZhjmLLzqpmbBim5UXjdhXTLGeap2Kq/flsAXcvZvmUL+2OL2R3PRyLAM8J2kL6OupiTrCaY0I5Ws+1/21cGnxpjE0RE53ViusiXdepXFV8Z6awcS3+gwjlZtUjZb58ObGc1cNr64P+AC/j/GBTD9aFfj0Or1/9XpjCEaGwxUT4QopApDT2hCUSeu+LTTEQEdEOREa0mEYu5qWYilWSnRCXJA6LqMf0fUS/9jU1oN+MmUDaJi0efWKSNMQnmseh8QhU9WT9Rol3no4M2GzjGT2HGtMnairGRO2SyzkXlxvS2+/wOiUfvoK3LnkSgcJxsCs+j+jp0e+UPOTvM+KUNHJIKqaRTTSPqIrCOmJ6qshzbMZYCQt8kkZsdZvdbVbzfEjvY8KyaJzAUrbVEA/1ugTQLGEr83hY0/mIP5GHyr42rNHMBxfrrrxg3j2iHItXNKoit+ksZ9d8rlQjTxt0MX8KjoC4E6KEyHyX4ucSRzVXbcJkHRZW8gi+RMigvkb+FvN5Nke0nhuwJ8GqGcg3F3oiXVmF3bmdHj2HwI5JPqO5MfRwLHpkjwR2jOzNo9gDd0y/TvHLho12jd5HZfJ69sgnzq4e4YY0486wVCRUPHOLlHUZ0Qz93hDfQb+kde8hKpzXPZINYaHK4gS6Ar3229RXOC4+fTOVI5tTetDhw44xsuWYaIU+hcT6PicvEtSPp1EzenqIWgSUHrRzfE4x9uETr+gEuk9p7EAPSxvU6/rinrZjWtHKsXHRnc5rNZB/XLuqM9a1slyASjV0Vm5D00Dc1R7zCrDd5ajZuhuqnpzAu2nNqiHsL7oG52jVPpYq1qe6Vg6ycQypKi5HsdnPJl+2UCc9rCXH/og4tiE10vQEtRlpTaul+tvRVRJiPWw8fZ0Tqt5Z22ltJiO6ryPlEZ+qc96TeOYlYVrIKRPRK8+3Ie5q+Xw34I6fgdou9wyVVZWNaZTNZtrlXYwk4qNGvOIKL0KsI/jy8exjlGYWxb4YvOAcImdsqpEWRXeQG82e74l16gXrlFXVePJKMyXG6SSi8a0Z5y3CZd5jaAmIwhWezeannWvwgChTHbER5vI6LGxw0fkccJo68xSHqqi0xsFV3a6V87IpbxS9XdaZ5XtVVtHbZZlygtx0dT9IgPWiQy7FOrCEX8rSvQ6pEHuCoiubf1jAvgB77UTnRawtfN4h6aPHKK9CXZWPOu1L0QVjHROW+fE1IniC/jfFCWjZOBay6VLRLOTOXiumhfzpkpEtJP0l41tIvuqQlNgb3BmNZZ501hRzPe7kUvnUtSdI2k3y84uqth6+hxRF3rMd3ZkMvd9wVqjdcRs9oLdQfz6/73Alb3bW8QTdd4g+FEOz01I7Ty8MrVjBid7L8908P4UnpT7vQmNxQs9wmjrVO0J+rtpAJFUsvqGe8ojOKjzeEV/QOWmTvh9Wdo9FUfm+MtF9p4y7Rcj3qO8/wP6wPG6kbxUblbtHVccuacg/U31LK7g9fZaeh7SM3mZktXrOuTN9XZM6tY2JS91kzjoQi3xoxmtf32Zk9Y9DiJN0cZ+5COTyXW0xvCJn5t+su+7782+hET1H9O2hMyaa9wGyzkOHlFSr6qasTiE75+78nL9bVBWcu9nNGzV9Vfz6LfIIt8AJcRQ86jaQQlrRyojz7qMRZFJ96kpwdytuqP3KjVPh1W1LSnnla7xt3aGk7rr8v9Zm/V+s84NnW/3NL/uDJ3fX79/V/3BdER+Jm+I2xemeuE+V+Zjiaq/8svLHyp8rf61laz+v/br2m2K9fEnLfCgqr7Xf/wHpIDV+</latexit><latexit sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">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</latexit>

EW [E[Y | T = t, W ]] = EW [E[ w W +T := ↵w W + ↵u U
Y :=

uU +

wW +

uU +

T

T | T = t, W ]]= EW [ w W + u E[U | T = t, W ] + t]✓◆t ↵w W= EW w W + u
+ t
↵u

u
u ↵w
= EW w W +
t
W+ t
↵u
↵u
u
u ↵w
= w E[W ] +t
E[W ] + t
↵
↵u
✓
◆ u✓
◆
↵
u
u w
=
+
t+
E[W ]
w
↵u
↵u

Brady Neal

Sensitivity Analysis: Linear Single Confounder

39 / 48

Bias in Simple Linear Setting Proof: Step 1<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:
<latexit sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">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</latexit><latexit sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">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</latexit><latexit sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">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</latexit>

EW [E[Y | T = t, W ]] = EW [E[ w W +T := ↵w W + ↵u U
Y :=

uU +

wW +

uU +

T

T | T = t, W ]]= EW [ w W + u E[U | T = t, W ] + t]✓◆t ↵w W= EW w W + u
+ t
↵u

u
u ↵w
= EW w W +
t
W+ t
↵u
↵u
u
u ↵w
= w E[W ] +t
E[W ] + t
↵
↵u
✓
◆ u✓
◆
↵
u
u w
=
+
t+
E[W ]
w
↵u
↵u

Brady Neal

Sensitivity Analysis: Linear Single Confounder

39 / 48

Bias in Simple Linear Setting Proof: Step 1<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:
<latexit sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">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</latexit><latexit sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">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</latexit><latexit sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">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</latexit>

EW [E[Y | T = t, W ]] = EW [E[ w W +T := ↵w W + ↵u U
Y :=

uU +

wW +

uU +

T

T | T = t, W ]]= EW [ w W + u E[U | T = t, W ] + t]✓◆t ↵w W= EW w W + u
+ t
↵u

u
u ↵w
= EW w W +
t
W+ t
↵u
↵u
u
u ↵w
= w E[W ] +t
E[W ] + t
↵
↵u
✓
◆ u✓
◆
↵
u
u w
=
+
t+
E[W ]
w
↵u
↵u

Brady Neal

Sensitivity Analysis: Linear Single Confounder

39 / 48

Bias in Simple Linear Setting Proof: Step 1<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:
<latexit sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">AAAZ+Xic1VhbbxtFFJ4WCG24teUp4mVpCiqqY5xQVB6IVClNhRCVWpReIBtF691xvMreurtOSFf+MYgXxCu/gld44t9wzjez3kvWu3YVkLBl7/jMOd+5zDlnZjyMPDdJB4O/L11+4823Vt6+cnX1nXffe/+Da9dvPEvCSWzLp3bohfGLoZVIzw3k09RNPfkiiqXlDz35fHi8w/PPT2ScuGGwl55F8sC3jgJ35NpWSqTD6ytfm7uHmWmHwSicBA5xTg3Tk6N03zB3981wktqhLw3Tdx3DTAk59WWQGtulHyeW1zPKCAeGGbtH4/TA+JT42uCHMrUOy7OEVYEy7hiaaRKEw0TGJ9KpszfOsKAjvdQqW/2aXpjmqkGvVmcuxBOKSLMzSxre4DvwC3/+fV8Y67ZhjmLLzqpmbBim5UXjdhXTLGeap2Kq/flsAXcvZvmUL+2OL2R3PRyLAM8J2kL6OupiTrCaY0I5Ws+1/21cGnxpjE0RE53ViusiXdepXFV8Z6awcS3+gwjlZtUjZb58ObGc1cNr64P+AC/j/GBTD9aFfj0Or1/9XpjCEaGwxUT4QopApDT2hCUSeu+LTTEQEdEOREa0mEYu5qWYilWSnRCXJA6LqMf0fUS/9jU1oN+MmUDaJi0efWKSNMQnmseh8QhU9WT9Rol3no4M2GzjGT2HGtMnairGRO2SyzkXlxvS2+/wOiUfvoK3LnkSgcJxsCs+j+jp0e+UPOTvM+KUNHJIKqaRTTSPqIrCOmJ6qshzbMZYCQt8kkZsdZvdbVbzfEjvY8KyaJzAUrbVEA/1ugTQLGEr83hY0/mIP5GHyr42rNHMBxfrrrxg3j2iHItXNKoit+ksZ9d8rlQjTxt0MX8KjoC4E6KEyHyX4ucSRzVXbcJkHRZW8gi+RMigvkb+FvN5Nke0nhuwJ8GqGcg3F3oiXVmF3bmdHj2HwI5JPqO5MfRwLHpkjwR2jOzNo9gDd0y/TvHLho12jd5HZfJ69sgnzq4e4YY0486wVCRUPHOLlHUZ0Qz93hDfQb+kde8hKpzXPZINYaHK4gS6Ar3229RXOC4+fTOVI5tTetDhw44xsuWYaIU+hcT6PicvEtSPp1EzenqIWgSUHrRzfE4x9uETr+gEuk9p7EAPSxvU6/rinrZjWtHKsXHRnc5rNZB/XLuqM9a1slyASjV0Vm5D00Dc1R7zCrDd5ajZuhuqnpzAu2nNqiHsL7oG52jVPpYq1qe6Vg6ycQypKi5HsdnPJl+2UCc9rCXH/og4tiE10vQEtRlpTaul+tvRVRJiPWw8fZ0Tqt5Z22ltJiO6ryPlEZ+qc96TeOYlYVrIKRPRK8+3Ie5q+Xw34I6fgdou9wyVVZWNaZTNZtrlXYwk4qNGvOIKL0KsI/jy8exjlGYWxb4YvOAcImdsqpEWRXeQG82e74l16gXrlFXVePJKMyXG6SSi8a0Z5y3CZd5jaAmIwhWezeannWvwgChTHbER5vI6LGxw0fkccJo68xSHqqi0xsFV3a6V87IpbxS9XdaZ5XtVVtHbZZlygtx0dT9IgPWiQy7FOrCEX8rSvQ6pEHuCoiubf1jAvgB77UTnRawtfN4h6aPHKK9CXZWPOu1L0QVjHROW+fE1IniC/jfFCWjZOBay6VLRLOTOXiumhfzpkpEtJP0l41tIvuqQlNgb3BmNZZ501hRzPe7kUvnUtSdI2k3y84uqth6+hxRF3rMd3ZkMvd9wVqjdcRs9oLdQfz6/73Alb3bW8QTdd4g+FEOz01I7Ty8MrVjBid7L8908P4UnpT7vQmNxQs9wmjrVO0J+rtpAJFUsvqGe8ojOKjzeEV/QOWmTvh9Wdo9FUfm+MtF9p4y7Rcj3qO8/wP6wPG6kbxUblbtHVccuacg/U31LK7g9fZaeh7SM3mZktXrOuTN9XZM6tY2JS91kzjoQi3xoxmtf32Zk9Y9DiJN0cZ+5COTyXW0xvCJn5t+su+7782+hET1H9O2hMyaa9wGyzkOHlFSr6qasTiE75+78nL9bVBWcu9nNGzV9Vfz6LfIIt8AJcRQ86jaQQlrRyojz7qMRZFJ96kpwdytuqP3KjVPh1W1LSnnla7xt3aGk7rr8v9Zm/V+s84NnW/3NL/uDJ3fX79/V/3BdER+Jm+I2xemeuE+V+Zjiaq/8svLHyp8rf61laz+v/br2m2K9fEnLfCgqr7Xf/wHpIDV+</latexit><latexit sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">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</latexit><latexit sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">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</latexit>

EW [E[Y | T = t, W ]] = EW [E[ w W +T := ↵w W + ↵u U
Y :=

uU +

wW +

uU +

<latexit sha1_base64="nsDHlNcfN/a+msRhqvjARAeDmuY=">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</latexit>

T

U=

T

↵w W
↵u

T | T = t, W ]]= EW [ w W + u E[U | T = t, W ] + t]✓◆t ↵w W= EW w W + u
+ t
↵u

u
u ↵w
= EW w W +
t
W+ t
↵u
↵u
u
u ↵w
= w E[W ] +t
E[W ] + t
↵
↵u
✓
◆ u✓
◆
↵
u
u w
=
+
t+
E[W ]
w
↵u
↵u

Brady Neal

Sensitivity Analysis: Linear Single Confounder

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Bias in Simple Linear Setting Proof: Step 1<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:
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sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">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</latexit>

EW [E[Y | T = t, W ]] = EW [E[ w W +T := ↵w W + ↵u U
Y :=

uU +

wW +

uU +

<latexit sha1_base64="nsDHlNcfN/a+msRhqvjARAeDmuY=">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</latexit>

T

U=

T

↵w W
↵u

T | T = t, W ]]= EW [ w W + u E[U | T = t, W ] + t]✓◆t ↵w W= EW w W + u
+ t
↵u

u
u ↵w
= EW w W +
t
W+ t
↵u
↵u
u
u ↵w
= w E[W ] +t
E[W ] + t
↵
↵u
✓
◆ u✓
◆
↵
u
u w
=
+
t+
E[W ]
w
↵u
↵u

Brady Neal

Sensitivity Analysis: Linear Single Confounder

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Bias in Simple Linear Setting Proof: Step 1<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:
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T := ↵w W + ↵u U
Y :=

wW +

uU +

<latexit sha1_base64="nsDHlNcfN/a+msRhqvjARAeDmuY=">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</latexit>

T

U=

T

↵w W
↵u

EW [E[Y | T = t, W ]] = EW [E[ w W + u U + T | T = t, W ]]= EW [ w W + u E[U | T = t, W ] + t]✓◆t ↵w W= EW w W + u
+ t
↵u

u
u ↵w
= EW w W +
t
W+ t
↵u
↵u
u
u ↵w
= w E[W ] +t
E[W ] + t
↵
↵u
✓
◆ u✓
◆
↵
u
u w
=
+
t+
E[W ]
w
↵u
↵u

Brady Neal

Sensitivity Analysis: Linear Single Confounder

39 / 48

Bias in Simple Linear Setting Proof: Step 1<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:
<latexit 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T := ↵w W + ↵u U
Y :=

wW +

uU +

<latexit sha1_base64="nsDHlNcfN/a+msRhqvjARAeDmuY=">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</latexit>

T

U=

T

↵w W
↵u

EW [E[Y | T = t, W ]] = EW [E[ w W + u U + T | T = t, W ]]= EW [ w W + u E[U | T = t, W ] + t]✓◆t ↵w W= EW w W + u
+ t
↵u

u
u ↵w
= EW w W +
t
W+ t
↵u
↵u
u
u ↵w
= w E[W ] +t
E[W ] + t
↵
↵u
✓
◆ u✓
◆
↵
u
u w
=
+
t+
E[W ]
w
↵u
↵u

Brady Neal

Sensitivity Analysis: Linear Single Confounder

39 / 48

Bias in Simple Linear Setting Proof: Step 1<latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

Assumed SCM:
<latexit 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sha1_base64="l6QovI303j1kFEIymG4MPpao3n8=">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</latexit>

T := ↵w W + ↵u U
Y :=

wW +

uU +

<latexit sha1_base64="nsDHlNcfN/a+msRhqvjARAeDmuY=">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</latexit>

T

U=

T

↵w W
↵u

EW [E[Y | T = t, W ]] = EW [E[ w W + u U + T | T = t, W ]]= EW [ w W + u E[U | T = t, W ] + t]✓◆t ↵w W= EW w W + u
+ t
↵u

u
u ↵w
= EW w W +
t
W+ t
↵u
↵u
u
u ↵w
= w E[W ] +t
E[W ] + t
↵
↵u
✓
◆ u✓
◆
↵
u
u w
=
+
t+
E[W ]
w
↵u
↵u

Brady Neal

Sensitivity Analysis: Linear Single Confounder

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Bias in Simple Linear Setting Proof: Step 2<latexit sha1_base64="NrWKbcmRTFqyoHCPV/d/zRNiLvA=">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</latexit>

Step 1:

Brady NealEW [E[Y | T = t, W ]] =

✓

+

u

↵u

◆

t+

✓

w

Sensitivity Analysis: Linear Single Confounder

u ↵w
↵u

◆

E[W ]

40 / 48

Bias in Simple Linear Setting Proof: Step 2<latexit sha1_base64="NrWKbcmRTFqyoHCPV/d/zRNiLvA=">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</latexit>

Step 1:EW [E[Y | T = t, W ]] =<latexit sha1_base64="N67bRDJCBPm8WdjeAKwYbuhq4NQ=">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</latexit>

EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] =

✓

✓

+

↵u

u

+
↵u
✓
+

= +

Brady Neal

u

◆

◆

t+

✓

w

✓

u ↵w
↵u

◆

E[W ]

◆

u ↵w
(1) +
E[W ]
w
↵u
◆
✓
◆
u
u ↵w
(0) +
E[W ]
w
↵u
↵u

u

↵u

Sensitivity Analysis: Linear Single Confounder

40 / 48

Bias in Simple Linear Setting Proof: Step 2<latexit sha1_base64="NrWKbcmRTFqyoHCPV/d/zRNiLvA=">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</latexit>

Step 1:EW [E[Y | T = t, W ]] =<latexit sha1_base64="N67bRDJCBPm8WdjeAKwYbuhq4NQ=">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</latexit><latexit sha1_base64="N67bRDJCBPm8WdjeAKwYbuhq4NQ=">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</latexit>

EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] =

✓

✓

+

↵u

u

+
↵u
✓
+

= +

Brady Neal

u

◆

◆

t+

✓

w

✓

u ↵w
↵u

◆

E[W ]

◆

u ↵w
(1) +
E[W ]
w
↵u
◆
✓
◆
u
u ↵w
(0) +
E[W ]
w
↵u
↵u

u

↵u

Sensitivity Analysis: Linear Single Confounder

40 / 48

Bias in Simple Linear Setting Proof: Step 2<latexit sha1_base64="NrWKbcmRTFqyoHCPV/d/zRNiLvA=">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</latexit>

Step 1:EW [E[Y | T = t, W ]] =<latexit sha1_base64="N67bRDJCBPm8WdjeAKwYbuhq4NQ=">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</latexit><latexit sha1_base64="N67bRDJCBPm8WdjeAKwYbuhq4NQ=">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</latexit><latexit sha1_base64="N67bRDJCBPm8WdjeAKwYbuhq4NQ=">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</latexit>

EW [E[Y | T = 1, W ]E[Y | T = 0, W ]] =

✓
✓

+

↵u

u

+
↵u
✓
+

= +

Brady Neal

u

◆
◆

t+

✓

w

✓

u ↵w
↵u

◆

E[W ]

◆

u ↵w
E[W ]
(1) +
w
↵u
◆
◆
✓
u ↵w
u
E[W ]
(0) +
w
↵u
↵u

u

↵u

Sensitivity Analysis: Linear Single Confounder

40 / 48

Bias in Simple Linear Setting Proof: Step 3

Brady Neal

Sensitivity Analysis: Linear Single Confounder

41 / 48

Bias in Simple Linear Setting Proof: Step 3
Bias = EW [E[Y | T = 1, W ]<latexit sha1_base64="GnN2BA26IUGztxfOS/e8BCHv0Sk=">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</latexit>

E[Y | T = 0, W ]]EW,U [E[Y | T = 1, W, U ]= +=Brady NealE[Y | T = 0, W, U ]]

u

↵u

u

↵u

Sensitivity Analysis: Linear Single Confounder

41 / 48

Bias in Simple Linear Setting Proof: Step 3
Bias = EW [E[Y | T = 1, W ] E[Y | T = 0, W ]]EW,U [E[Y | T = 1, W, U ] E[Y | T = 0, W, U ]]<latexit sha1_base64="GnN2BA26IUGztxfOS/e8BCHv0Sk=">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</latexit><latexit sha1_base64="GnN2BA26IUGztxfOS/e8BCHv0Sk=">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</latexit>= +=Brady Neal

u

↵u

u

↵u

Sensitivity Analysis: Linear Single Confounder

41 / 48

Bias in Simple Linear Setting Proof: Step 3
Bias = EW [E[Y | T = 1, W ] E[Y | T = 0, W ]]EW,U [E[Y | T = 1, W, U ] E[Y | T = 0, W, U ]]<latexit sha1_base64="GnN2BA26IUGztxfOS/e8BCHv0Sk=">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</latexit><latexit 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+=Brady Neal

u

↵u

u

↵u

Sensitivity Analysis: Linear Single Confounder

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Bias in Simple Linear Setting Proof: Step 3<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit>

W

Bias = EW [E[Y | T = 1, W ] E[Y | T = 0, W ]]EW,U [E[Y | T = 1, W, U ] E[Y | T = 0, W, U ]]<latexit 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u

u

↵u

T

Y

u

↵u

<latexit sha1_base64="oLNh5jgGL8lSG3RxxH8XLmVD2Fg=">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</latexit>

T := ↵w W + ↵u U

Y :=Brady Neal↵u

Sensitivity Analysis: Linear Single Confounder

wW +

uU +

T

41 / 48

Generalization to Arbitrary Linear SCMs
<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit>

We’ve considered specifically the ATE in this simple graph
E[Y (1)

W

U

T

Y

<latexit sha1_base64="DzOYVEAuC43gMru9XrTTWtcmBas=">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</latexit>

Brady Neal

Y (0)]

Sensitivity Analysis: Linear Single Confounder

42 / 48

Generalization to Arbitrary Linear SCMs
<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit>

We’ve considered specifically the ATE in this simple graph
E[Y (1)

W

U

T

Y

<latexit sha1_base64="DzOYVEAuC43gMru9XrTTWtcmBas=">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</latexit>

Y (0)]

See “Sensitivity Analysis of Linear Structural Causal Models” from Cinelli
et al. (2019) for arbitrary estimands in arbitrary graphs, where thestructural equations are still linear
Brady Neal

Sensitivity Analysis: Linear Single Confounder

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<latexit sha1_base64="8wHEjXx9bhbrxOrwWayhWz3nk5g=">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</latexit>

SCM:

T := ↵w W + ↵u U

Y :=

wW +

uU +

T

Questions:
1. Given the above SCM, show that
EW,U [E[Y | T = 1, W, U ]<latexit sha1_base64="W/TpkSuEvYHYjtBXNqXDnMS7kjc=">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</latexit>

E[Y | T = 0, W, U ]] =2. Does what we have shown in this section work if W is avector?
3. How about if U is a vector?

Bounds
No-Assumptions Bound
Monotone Treatment Response
Monotone Treatment Selection
Optimal Treatment Selection

Sensitivity Analysis
Linear Single Confounder
Towards More General Settings

Brady Neal

Sensitivity Analysis: Towards More General Settings

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Binary Treatment

Brady Neal

Sensitivity Analysis: Towards More General Settings

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Binary Treatment<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit><latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

T := ↵w W + ↵u U
Y :=

wW +

Brady Neal

uU +

T

W

U

T

Y

Sensitivity Analysis: Towards More General Settings

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Binary Treatment<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">AAAWTHicrVjbbttGEN3YvSTuLUnhvrQPbOwUCSCrstsiAQIDAZwERdEUSeFcWssIKHIlEeKt5MqqI+gb+md9738UfSkKdObsUiQlipSDSLC4nJ05Mzu33XUv9r1UdTp/XdrYfOfd996/fGXrgw8/+viTq9euP0+jceLIZ07kR8nLnp1K3wvlM+UpX76ME2kHPV++6I2OeP7FmUxSLwqP1XksTwN7EHp9z7EVkV5d2/ij25MDL5wqb/Q69hw1TuRsyzKfm9ZPkSvT+Xs3pNeTVNlKnlq3uk4U9qNx6BL+bctW1q1Oq3Pbmu4WJ3Zn96rEW66dDqXLKOMw6qUyOZNuCe8k8QZDZR1GfauId8r4lSIrNLEKRS5RgQwVwfakH02qYedsNVDRWDlRIMtAlfYA0rBngAXPPnQHRc/GthouulQSS8n6e2vxZyYuc69w9Rp66iVXayx4vpK3K0O3lHuvru502h18rOXBvhnsCPN5El278rPoCldEwhFjEQgpQqFo7AtbpPQ9EfuiI2KinYop0RIaeZiXYia2SHZMXJI4bKKO6HdAbyeGGtI7Y6aQdkiLT38JSVripuFxadwHVT9Zv1XgXaVjCmy28ZyePYMZEFWJIVGb5DLO9eV69A0aVq1oDXexWo9WEoPCfnBKa+7T06d3RSvk33PilDRySSqhkUM0n6iawjoSemrPs2+GiIQNPkkjtrrO7jqreT6i74iwbBqnsJRttcQjE5cQmiVsZR4fMV2N+DutUNtXh9Wfr8FD3PUqmPeYKCPxmkZl5DqdxexazaUM8qxCF/MrcITEnRIlQuZ75D+POMq56hAm67ARyQHWEiOD2gb5B8xn2RxTPPdgT4qoWcg3D3piU1m53ZmdPj17wE5IfkpzQ+hhX7TIHgnsBNmbebEF7oTeJnhzYKOzQG+jMjmeLVoTZ1eLcCOa8eZY2hPan5lF2rop0Szz3RM/Qr+kuLfgFc7rFslGsFBncQpdoYn9IfUV9ktAv0xlz2aUFnQEsGOIbBkRLdenkVjf17SKFPXjG9QpPX14LQZKC9rZPxOMA6yJIzqG7gmNXehhaYt6XVvcMXbMSlrZNx6607JWC/nHtas746JWlgtRqZbJykNo6ohvzYo5Amx30WuO6Ya6J6dY3WzBqh7sz7sG52jZPpbK41OOlYtsHEKqjMterF5n1VoOUCctxJJ9PyCOQ0j1DT1FbcZG01ah/o5MlUSIh4NnYHJC1ztrmyzMTIkeGE/5xKfrnPcknvmNMG3kVBfeK87XIT408tluwB1/Cmq93HNUVlk2odF0PlMv72Ek4R894ohrvBi+jrGWL+d/VmFmXey3gxcuIXLGKoO0LrqL3Khe+bHYoV6wQ1lV9idHmikJTicxjXfnnLuEy7wjaAmJwhU+nc/PGmPwgCgz47E+5rI6zG3w0PlccHZN5mkOXVFqgYOrul4r52VV3mh6vaw7z/eyrKbXyzLlDLnpmX6QAutlg5xCHFgiKGTpcYNUhD1B07XNv6xhX4i9dmzyIjEWvmiQDNBj9KoiU5WPG+1T6IKJ8QnL/PoGHjxD/5vhBHRRP+ay6kLezOXO38inufzkgp7NJYML+jeXfN0gKbE3eHMayzxtrCnmetLIpfOpaU+QtJtk5xddbS389siLvGe7pjNZZr/hrNC74yF6QGut/ry873Al7zfW8Rjdt4c+lECzW1M7z94aWh7BsdnLs908O4WnhT7vQWN+Qp/iNDUxO0J2rtqDJ7Uvvqee8pjOKjw+Et/QOWmffh+Vdo91Ufm+MjZ9p4h7QMh3qO8/wP5wcdzY3Cr2SnePso6HpCH7m5lbWs7tm7P0KqSL6K1G1tFzl870i5r0qW1IXPomc96AmOdDNV59fKuR9X8cIpyk8/vM20Au3tXWw8tzZvXNuum+v/oWGtOzT78+OmNqeB8g63x0SEm1qm/K+hRytHTn5/w9oKrg3J3euL6gr4y/eIsc4BY4Jo6cR98GFKQ1rYi46j4aQ0aZU1eKu1t+Q22Xbpwab9G2tJBXgcE7NB1Kmq679erqzv7if7GWB88P2vvftTtPD3bu3zX/4bosPhc3xC3y0x1xnyrzCfnV2fhn84vNm5tfbf+5/ff2v9v/adaNS0bmU1H6fPbe/3rTp4E=</latexit><latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

T := ↵w W + ↵u U
Y :=

wW +

uU +

<latexit sha1_base64="NBrqW75gJnWZuTZgPE4Oz3kAdVY=">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</latexit>

T

W

U

P (T = 1 | W, U ) := sigmoid (↵w W + ↵u U )
Y :=

wW +

uU +

T +N

1
where sigmoid(x) =1+e x<latexit sha1_base64="zSoSZLwk8CKCkiHoyKEiEx78wYA=">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</latexit>

T

Y

Rosenbaum & Rubin (1983) and Imbens (2003)

Brady Neal

Sensitivity Analysis: Towards More General Settings

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Binary Treatment<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit><latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

T := ↵w W + ↵u U
Y :=

wW +

uU +

<latexit sha1_base64="NBrqW75gJnWZuTZgPE4Oz3kAdVY=">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</latexit>

T

W

U

P (T = 1 | W, U ) := sigmoid (↵w W + ↵u U )
Y :=

wW +

uU +

T +N

1
where sigmoid(x) =1+e x<latexit sha1_base64="zSoSZLwk8CKCkiHoyKEiEx78wYA=">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</latexit>

T

Y

Rosenbaum & Rubin (1983) and Imbens (2003)• Simple parametric form for T

Brady Neal

Sensitivity Analysis: Towards More General Settings

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Binary Treatment<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit><latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

T := ↵w W + ↵u U
Y :=

wW +

uU +

<latexit sha1_base64="NBrqW75gJnWZuTZgPE4Oz3kAdVY=">AAAVAXicrVhbj9tEFJ6UW7vc2rJvvJhuQa2aDclSVIQUqdJuK4QoatG23VJXK8eeJNb6hj3Z7TbKE78G8YJ45VfwCPwZzvlmHNu5OamaKPHkzDnfOXNuM5NeEviZarf/aVx46+133n3v4qWt9z/48KOPL1+5+iSLR6krH7txEKdHPSeTgR/Jx8pXgTxKUumEvUA+7Z3s8/zTU5lmfhwdqvNEvgidQeT3fddRRDq+0jiwkxu2IhEVykhZXatj2aHvWbYbR/14FHkk3LTsURT3MpmeSq9Ev2l98W3XspV8qcaZPwhj35tYdiD76oZlO0EydI7LMKdOUIG1bk25FsKDf+GMZaf+YKhuWra9ZdHLjkfKjUOp7elJVa9YM22qlwQ9GSjHKrnslvXj8eWddquNlzU/6JjBjjCvh/GVSz8JW3giFq4YiVBIEQlF40A4IqP3c9ERbZEQ7YUYEy2lkY95KSZii2RHxCWJwyHqCX0P6NdzQ43oN2NmkHZJS0CflCQt8bnh8WjcB1U/Wb9V4l2mYwxstvGcnj2DGRJViSFR6+RyzvXlevQOa1ataA3fYLU+rSQBhf3gVtbcp2dAvxWtkL/PiVPSyCOplEYu0QKiagrrSOmpPc++GSISDvgkjdjqVXavsprnY3qfEJZD4wyWsq2WuG/iEkGzhK3MEyCmyxFf0gq1fauw+tM1+Ii7XgXzHhLlRLyiURV5lc5ydi3nUgZ5skAX8ytwRMSdESVG5vvkP584qrnqEibrcBDJAdaSIINaBvl7zOfZnFA8d2FPhqhZyDcfehJTWYXduZ0BPXvATkl+THND6GFfNMkeCewU2Zt7sQnulH6d4ZcLG90ZeguVyfFs0po4u5qEG9OMP8XSntD+zC3S1o2JZpn3rvgB+iXFvQmvcF43STaGhTqLM+iKTOy71FfYLyF9M5U9m1Oa0BHCjiGy5YRohT6NxPq+pFVkqJ/AoI7pGcBrCVCa0M7+OcM4xJo4oiPoPqOxBz0sbVGva4k7xo5JRSv7xkd3mtdqIf+4dnVnnNXKchEq1TJZ2YWmtrhtVswRYLvLXnNNN9Q9OcPqJjNW9WB/0TU4R6v2sVQRn2qsPGTjEFJVXPbi4nUuWsse6qSJWLLvB8TRhVTf0DPUZmI0bZXqb99USYx4uHiGJid0vbO2s5mZMdFD46mA+HSd857EM78QpoOcsuG98vwqxHtGPt8NuOOPQV0t9wSVVZVNaTSezqyW9zGS8I8eccQ1XgJfJ1jLZ9OPVZpZF/vN4EVziJyxyiCti+4hNxav/FDsUC/Yoayq+pMjzZQUp5OExtennNcJl3lPoCUiClf4eDo/qY3BAVEmxmN9zOV1WNjgo/N54LRN5mkOXVFqhoOrerVWzstFeaPpq2W9ab5XZTV9tSxTTpGbvukHGbCOauQU4sASYSlLD2ukYuwJmq5tfraGfRH22pHJi9RY+LRGMkSP0auKTVU+qLVPoQumxics8/NrePAU/W+CE9Cmfixk1UbeLOTOX8unhfzZhp4tJMMN/VtIvqqRlNgb/CmNZR7V1hRzPazl0vlUtydI2k3y84uutia+e+RF3rM905kss99wVujdsYse0FyrP8/vO1zJndo6HqH79tCHUmj2VtTO4zeGVkRwZPbyfDfPT+FZqc/70Fic0Mc4TZ2ZHSE/V+3Ck9oX31FPeUBnFR7vi6/onNSh7/uV3WNdVL6vjEzfKePuEfId6vsH2B82x03MrWK3cveo6rhHGvLPxNzSCu7AnKWXIW2idzGyjp43d6af1aRPbUPi0jeZ8xrEIh8W462O72Jk/Y9DjJN0cZ95E8jlu9p6eEXOLL9Z1933l99CE3r26TtAZ8wM7wGyLkCHlFSr+qasTyH7c3d+zt89qgrO3fG1qzP6qvizt8gBboEj4ih49G1AQVrTyojL7qMJZJQ5dWW4uxU31FblxqnxZm3LSnkVGryu6VDSdN2t48s7ndl/seYHT/Zana9b7Ue3d+7eNv9wXRSfimviBvnpjrhLlfmQ/Oo2fm/83fi38d/2r9u/bf+x/admvdAwMp+Iymv7r/8BuS9KiA==</latexit>

T

W

U

P (T = 1 | W, U ) := sigmoid (↵w W + ↵u U )
Y :=

wW +

uU +

T +N

1
where sigmoid(x) =1+e x<latexit sha1_base64="zSoSZLwk8CKCkiHoyKEiEx78wYA=">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</latexit>

T

Y

Rosenbaum & Rubin (1983) and Imbens (2003)• Simple parametric form for T• Simple parametric form for Y

Brady Neal

Sensitivity Analysis: Towards More General Settings

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Binary Treatment<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit><latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">AAAU4HicrVhbbxtFFB6XWxtuSckbL0tdEBK2sUNQUKVIlZJWCFEpRblBXUV7Gccr762764TU8jviBfHKr+AV/gn/hnO+mfXu+rZ2FVv2js+c850z5zYztiLPTdJ2+7/anbfefufd9+7e23j/gw8/+nhz6/5pEg5jW57YoRfG55aZSM8N5Enqpp48j2Jp+pYnz6zBAc+fXck4ccPgOL2J5EvfvAzcnmubKZEutmr1bkr8qS+D1Pji0b7RNb2ob1507TDohcPAIdkr0zOKv42vJlzDILQSGV9JZ5p/7ozR9UxLeiP56tFEa5NNN+NmLtDMJNzgcmx0uxsGvbrhMLVDXyojLZlW26iY1jWRBB3ppaaRO+Zis95utfEyZgcdPagL/ToKt+79JLrCEaGwxVD4QopApDT2hCkSer8QHdEWEdFeihHRYhq5mJdiLDZIdkhckjhMog7o+5J+vdDUgH4zZgJpm7R49IlJ0hCfax6Hxj1Q1ZP1GwXeRTpGwGYbb+hpaUyfqKnoE7VKLuNcXc6it1+x6pTW8B1W69JKIlDYD3ZpzT16evQ7pRXy9w1xSho5JBXTyCaaR1RFYR0xPZXn2Td9RMIEn6QRW73M7mVW83xI7wFhmTROYCnbaoinOi4BNEvYyjweYroY8VdaobJvGVZvsgYXcVerYN5jogzEaxqVkZfpLGbXYq5UI4/n6GL+FBwBcSdECZH5LvnPJY5yrtqEyTpMRPISa4mQQS2N/APms2yOKJ5N2JMgagbyzYWeSFdWbndmp0dPC9gxyY9org897IsG2SOBHSN7My82wB3Tr2v8smGjPUVvoTI5ng1aE2dXg3BDmnEnWMoTyp+ZRcq6EdEM/W6KH6FfUtwb8ArndYNkQ1iosjiBrkDHfp/6CvvFp2+msmczSgM6fNjRR7YMiJbrU0is72taRYL68TTqiJ4evBYBpQHt7J9rjH2siSM6hO5rGjvQw9IG9bqW2NN2jEta2TcuutOsVgP5x7WrOuO0VpYLUKmGzsp9aGqLXb1ijgDbXfSarbuh6skJVjeessqC/XnX4Bwt28dSeXzKsXKQjX1IlXHZi/PXOW8tO6iTBmLJvr8kjn1I9TQ9QW1GWtNGof4OdJWEiIeNp69zQtU7a7uemhkR3dee8ohP1TnvSTzzijBN5FQX3ivOL0N8ouWz3YA7/gjU5XKnqKyybEyj0WRmubyLkYR/1IgjrvAi+DrCWj6bfIzCzKrYt4MXzCByxqYaaVV0B7kxf+XHok69oE5ZVfYnR5opMU4nEY0fTjgfEi7zDqAlIApX+GgyP66MwSFRxtpjPcxldZjb4KLzOeDs6sxTHKqi0ikOrurlWjkv5+WNoi+XdSb5XpZV9OWyTLlCbrq6HyTAOq+QSxEHlvALWXpcIRViT1B0ZfPPK9gXYK8d6ryItYVnFZI+eoxaVair8lmlfSm6YKx9wjK/vIEHr9D/xjgBrevHXDZdy5u53M0b+TSXv17Ts7mkv6Z/c8nXFZISe4M7obHM88qaYq6jSi6VT1V7gqTdJDu/qGpr4NsiL/Ke7ejOZOj9hrNC7Y776AGNlfrz7L7DldyprOMhuq+FPhRDs7Okdk5uDS2P4FDv5dlunp3Ck0Kfd6ExP6GPcJq61jtCdq5qwpPKF99TT3lGZxUeH4hv6JzUoe+npd1jVVS+rwx13yni7hDyHvX9Q+wP6+NG+lbRLN09yjqekIbsM9a3tJzb02fpRUjr6J2PrKLnzJzppzWpU1ufuNRN5qYCMc+H+XjL4zsfWf3jEOIknd9nbgO5eFdbDS/PmcU366r7/uJbaETPHn176IyJ5j1E1nnokJJqVd2U1SnkYObOz/m7Q1XBuTt6cH9KXxl/+hZ5iVvgkDhyHnUbSCGtaEXERffRCDKpPnUluLvlN9RW6cap8KZtSwp55Wu8fd2hpO66Gxeb9c70v1izg9OdVufbVvv5bv3xrv6H6674VDwQX5Kf9sRjqswj8qtd+732d+2f2r/b1vZv239s/6lY79S0zCei9Nr+638pMT9R</latexit>

T := ↵w W + ↵u U
Y :=

wW +

uU +

<latexit sha1_base64="NBrqW75gJnWZuTZgPE4Oz3kAdVY=">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</latexit>

T

W

U

P (T = 1 | W, U ) := sigmoid (↵w W + ↵u U )
Y :=

wW +

uU +

T +N

1
where sigmoid(x) =1+e x<latexit sha1_base64="zSoSZLwk8CKCkiHoyKEiEx78wYA=">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</latexit>

T

Y

Rosenbaum & Rubin (1983) and Imbens (2003)• Simple parametric form for T• Simple parametric form for Y• U is binary

Brady Neal

Sensitivity Analysis: Towards More General Settings

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Binary Treatment<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">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</latexit><latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

T := ↵w W + ↵u U
Y :=

wW +

uU +

<latexit sha1_base64="NBrqW75gJnWZuTZgPE4Oz3kAdVY=">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</latexit>

T

W

U

P (T = 1 | W, U ) := sigmoid (↵w W + ↵u U )
Y :=

wW +

uU +

T +N

1
where sigmoid(x) =1+e x<latexit sha1_base64="zSoSZLwk8CKCkiHoyKEiEx78wYA=">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</latexit>

T

Y

Rosenbaum & Rubin (1983) and Imbens (2003)• Simple parametric form for T• Simple parametric form for Y• U is binary• U is a scalar (only one unobserved confounder)
Brady Neal

Sensitivity Analysis: Towards More General Settings

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Binary Treatment<latexit sha1_base64="2mDnz8YUnG1kBWtB9nKsMPMaveA=">AAAWTHicrVjbbttGEN3YvSTuLUnhvrQPbOwUCSCrstsiAQIDAZwERdEUSeFcWssIKHIlEeKt5MqqI+gb+md9738UfSkKdObsUiQlipSDSLC4nJ05Mzu33XUv9r1UdTp/XdrYfOfd996/fGXrgw8/+viTq9euP0+jceLIZ07kR8nLnp1K3wvlM+UpX76ME2kHPV++6I2OeP7FmUxSLwqP1XksTwN7EHp9z7EVkV5d2/ij25MDL5wqb/Q69hw1TuRsyzKfm9ZPkSvT+Xs3pNeTVNlKnlq3uk4U9qNx6BL+bctW1q1Oq3Pbmu4WJ3Zn96rEW66dDqXLKOMw6qUyOZNuCe8k8QZDZR1GfauId8r4lSIrNLEKRS5RgQwVwfakH02qYedsNVDRWDlRIMtAlfYA0rBngAXPPnQHRc/GthouulQSS8n6e2vxZyYuc69w9Rp66iVXayx4vpK3K0O3lHuvru502h18rOXBvhnsCPN5El278rPoCldEwhFjEQgpQqFo7AtbpPQ9EfuiI2KinYop0RIaeZiXYia2SHZMXJI4bKKO6HdAbyeGGtI7Y6aQdkiLT38JSVripuFxadwHVT9Zv1XgXaVjCmy28ZyePYMZEFWJIVGb5DLO9eV69A0aVq1oDXexWo9WEoPCfnBKa+7T06d3RSvk33PilDRySSqhkUM0n6iawjoSemrPs2+GiIQNPkkjtrrO7jqreT6i74iwbBqnsJRttcQjE5cQmiVsZR4fMV2N+DutUNtXh9Wfr8FD3PUqmPeYKCPxmkZl5DqdxexazaUM8qxCF/MrcITEnRIlQuZ75D+POMq56hAm67ARyQHWEiOD2gb5B8xn2RxTPPdgT4qoWcg3D3piU1m53ZmdPj17wE5IfkpzQ+hhX7TIHgnsBNmbebEF7oTeJnhzYKOzQG+jMjmeLVoTZ1eLcCOa8eZY2hPan5lF2rop0Szz3RM/Qr+kuLfgFc7rFslGsFBncQpdoYn9IfUV9ktAv0xlz2aUFnQEsGOIbBkRLdenkVjf17SKFPXjG9QpPX14LQZKC9rZPxOMA6yJIzqG7gmNXehhaYt6XVvcMXbMSlrZNx6607JWC/nHtas746JWlgtRqZbJykNo6ohvzYo5Amx30WuO6Ya6J6dY3WzBqh7sz7sG52jZPpbK41OOlYtsHEKqjMterF5n1VoOUCctxJJ9PyCOQ0j1DT1FbcZG01ah/o5MlUSIh4NnYHJC1ztrmyzMTIkeGE/5xKfrnPcknvmNMG3kVBfeK87XIT408tluwB1/Cmq93HNUVlk2odF0PlMv72Ek4R894ohrvBi+jrGWL+d/VmFmXey3gxcuIXLGKoO0LrqL3Khe+bHYoV6wQ1lV9idHmikJTicxjXfnnLuEy7wjaAmJwhU+nc/PGmPwgCgz47E+5rI6zG3w0PlccHZN5mkOXVFqgYOrul4r52VV3mh6vaw7z/eyrKbXyzLlDLnpmX6QAutlg5xCHFgiKGTpcYNUhD1B07XNv6xhX4i9dmzyIjEWvmiQDNBj9KoiU5WPG+1T6IKJ8QnL/PoGHjxD/5vhBHRRP+ay6kLezOXO38inufzkgp7NJYML+jeXfN0gKbE3eHMayzxtrCnmetLIpfOpaU+QtJtk5xddbS389siLvGe7pjNZZr/hrNC74yF6QGut/ry873Al7zfW8Rjdt4c+lECzW1M7z94aWh7BsdnLs908O4WnhT7vQWN+Qp/iNDUxO0J2rtqDJ7Uvvqee8pjOKjw+Et/QOWmffh+Vdo91Ufm+MjZ9p4h7QMh3qO8/wP5wcdzY3Cr2SnePso6HpCH7m5lbWs7tm7P0KqSL6K1G1tFzl870i5r0qW1IXPomc96AmOdDNV59fKuR9X8cIpyk8/vM20Au3tXWw8tzZvXNuum+v/oWGtOzT78+OmNqeB8g63x0SEm1qm/K+hRytHTn5/w9oKrg3J3euL6gr4y/eIsc4BY4Jo6cR98GFKQ1rYi46j4aQ0aZU1eKu1t+Q22Xbpwab9G2tJBXgcE7NB1Kmq679erqzv7if7GWB88P2vvftTtPD3bu3zX/4bosPhc3xC3y0x1xnyrzCfnV2fhn84vNm5tfbf+5/ff2v9v/adaNS0bmU1H6fPbe/3rTp4E=</latexit><latexit sha1_base64="GaCQweg2FtFb74Cd3rmFeXbcwn8=">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</latexit>

T := ↵w W + ↵u U
Y :=

wW +

uU +

<latexit sha1_base64="NBrqW75gJnWZuTZgPE4Oz3kAdVY=">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</latexit>

T

W

U

P (T = 1 | W, U ) := sigmoid (↵w W + ↵u U )
Y :=

wW +

uU +

T +N

1
where sigmoid(x) =1+e x<latexit sha1_base64="zSoSZLwk8CKCkiHoyKEiEx78wYA=">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</latexit>

T

Cinelli & Hazlett (2020) drop
many of these assumptions

Brady NealY

Rosenbaum & Rubin (1983) and Imbens (2003)• Simple parametric form for T• Simple parametric form for Y• U is binary• U is a scalar (only one unobserved confounder)

Sensitivity Analysis: Towards More General Settings

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Making Sense of Sensitivity: Extending Omitted
Variable Bias Cinelli & Hazlett (2020)

Brady Neal

Sensitivity Analysis: Towards More General Settings46 / 48

Making Sense of Sensitivity: Extending Omitted
Variable Bias Cinelli & Hazlett (2020)• Assumes no parametric form
for structural equation for T

Brady Neal

Sensitivity Analysis: Towards More General Settings46 / 48

Making Sense of Sensitivity: Extending Omitted
Variable Bias Cinelli & Hazlett (2020)• Assumes no parametric form
for structural equation for T• Allows for multiple confounders(U can be a vector)

Brady Neal

Sensitivity Analysis: Towards More General Settings46 / 48

Making Sense of Sensitivity: Extending Omitted
Variable Bias Cinelli & Hazlett (2020)• Assumes no parametric form
for structural equation for T• Allows for multiple confounders(U can be a vector)• No assumed distribution on U

Brady Neal

Sensitivity Analysis: Towards More General Settings46 / 48

Making Sense of Sensitivity: Extending Omitted
Variable Bias Cinelli & Hazlett (2020)• Assumes no parametric form
for structural equation for T• Allows for multiple confounders(U can be a vector)• No assumed distribution on U
• Gives rigorous bound to tells us
when we can know that ourcausal effect estimate issufficiently robust to
unobserved confounding
Brady Neal

Sensitivity Analysis: Towards More General Settings46 / 48

Making Sense of Sensitivity: Extending Omitted
Variable Bias Cinelli & Hazlett (2020)• Assumes no parametric form
for structural equation for T• Allows for multiple confounders(U can be a vector)• No assumed distribution on U
• Gives rigorous bound to tells us
when we can know that ourcausal effect estimate issufficiently robust to
unobserved confounding
Figure 4:
Brady Neal

Sensitivity Analysis: Towards More General Settings46 / 48

Making Sense of Sensitivity: Extending Omitted
Variable Bias Cinelli & Hazlett (2020)• Assumes no parametric form
for structural equation for T• Allows for multiple confounders(U can be a vector)• No assumed distribution on U
• Gives rigorous bound to tells us
when we can know that ourcausal effect estimate is Imbens (2003)
and follow-ups
sufficiently robust to
unobserved confounding
Figure 4:
Brady Neal

Sensitivity Analysis: Towards More General Settings46 / 48

Sense and Sensitivity Analysis Veitch & Zaveri (2020)

Brady Neal

Sensitivity Analysis: Towards More General Settings

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Sense and Sensitivity Analysis Veitch & Zaveri (2020)
Gives the same nice sensitivity curves we’ve been seeing, but with more
flexible parameterization

Brady Neal

Sensitivity Analysis: Towards More General Settings

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Sense and Sensitivity Analysis Veitch & Zaveri (2020)
Gives the same nice sensitivity curves we’ve been seeing, but with more
flexible parameterization
Both the treatment mechanism and the outcome mechanism can bemodeled with arbitrary machine learning models, and we still get aclosed-form expression for the bias (assuming well-specification)

Brady Neal

Sensitivity Analysis: Towards More General Settings

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Lots of Other Sensitivity Analysis Methods• An Introduction to Sensitivity Analysis for Unobserved Confounding in
Non-Experimental Prevention Research (Liu, Kuramoto, & Stuart., 2013)• Rosenbaum has several (Rosenbaum 2002, 2010, 2017)• Unmeasured Confounding for General Outcomes, Treatments, andConfounders (VanderWeele & Arah, 2011)• Sensitivity Analysis Without Assumptions (Ding & VanderWeele, 2018)• Flexible sensitivity analysis for observational studies without observable
implications (Franks, D'Amour, & Feller, 2019)• Bounds on the conditional and average treatment effect with unobserved
confounding factors (Yadlowsky et al., 2018)
Brady Neal

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