Identification
Brady Nealcausalcourse.com

The magic of randomized experiments
Frontdoor adjustment
Pearl’s do-calculus
Determining identifiability from the graph

Brady Neal

2 / 40

The magic of randomized experiments
Frontdoor adjustment
Pearl’s do-calculus
Determining identifiability from the graph

Brady Neal

The magic of randomized experiments

3 / 40Randomized experiments are magic.

Randomized experiments are magic.
No unobserved confounding

Randomized control trial (RCT)T

Brady Neal

The magic of randomized experiments

5 / 40Randomized control trial (RCT)T
Went to sleep with shoes on(T = 1)
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Brady Neal

drunk
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Went to sleep without shoes on(T = 0)sober sober
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The magic of randomized experiments

5 / 40Randomized control trial (RCT)T
Slept with shoes on(T = 1)
drunk

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drunk
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sober

Brady Neal

drunk
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Went to sleep without shoes on(T = 0)sober sober
sober sober
drunk
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The magic of randomized experiments

5 / 40Randomized control trial (RCT)T
Slept with shoes on(T = 1)
drunk

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sober

Brady Neal

drunk
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Slept without shoes on(T = 0)sober sober
sober sober
drunk
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drunk
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The magic of randomized experiments

5 / 40Randomized control trial (RCT)T
Slept with shoes on(T = 1)

Slept without shoes on(T = 0)
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sober

sober
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Brady Neal

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The magic of randomized experiments

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5 / 40Few different perspectives on the magic
Comparability and covariate balance
Exchangeability
No backdoor paths

Brady Neal

The magic of randomized experiments

6 / 40Comparability and covariate balance: intuition

Brady Neal

The magic of randomized experiments

7 / 40Comparability and covariate balance: intuition
Treatment and control groups are the same in all aspects except treatment

Brady Neal

The magic of randomized experiments

7 / 40Comparability and covariate balance: intuition
Treatment and control groups are the same in all aspects except treatment

Brady Neal

The magic of randomized experiments

7 / 40Comparability and covariate balance: intuition
Treatment and control groups are the same in all aspects except treatment(T = 1)

drunk
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sober

Brady Neal

drunk
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(T = 0)sober sober
sober sober
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The magic of randomized experiments

7 / 40Comparability and covariate balance: intuition
Treatment and control groups are the same in all aspects except treatment(T = 1)

drunk

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Brady Neal

drunk

sober

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(T = 0)
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The magic of randomized experiments

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7 / 40Covariate balance definition

Brady Neal

The magic of randomized experiments

8 / 40Covariate balance definition
We have covariate balance if the distribution of covariates X is the sameacross treatment groups. More formally,

Brady Neal

The magic of randomized experiments

8 / 40Covariate balance definition
We have covariate balance if the distribution of covariates X is the sameacross treatment groups. More formally,dP (X | T = 1) = P (X | T = 0)

Brady Neal

The magic of randomized experiments

8 / 40Randomization implies covariate balance

Brady Neal

The magic of randomized experiments

9 / 40Randomization implies covariate balance
Because T is not at all determined by X (solely by a coin flip), T ?? X

Brady Neal

The magic of randomized experiments

9 / 40Randomization implies covariate balance
Because T is not at all determined by X (solely by a coin flip), T ?? X
d

P (X | T = 1) = P (X)

Brady Neal

The magic of randomized experiments

9 / 40Randomization implies covariate balance
Because T is not at all determined by X (solely by a coin flip), T ?? X
d

P (X | T = 1) = P (X)d

P (X | T = 0) = P (X)

Brady Neal

The magic of randomized experiments

9 / 40Randomization implies covariate balance
Because T is not at all determined by X (solely by a coin flip), T ?? X
d

P (X | T = 1) = P (X)d

P (X | T = 0) = P (X)d

P (X | T = 1) = P (X | T = 0)

Brady Neal

The magic of randomized experiments

9 / 40Covariate balance implies association is causationd

P (X | T = 1) = P (X | T = 0) =) T ?? X

Brady Neal

The magic of randomized experiments

10 / 40

Covariate balance implies association is causationd

P (X | T = 1) = P (X | T = 0) =) T ?? X

P (y | do(t)) ==Xx

P (y | t, x)P (x)X P (y | t, x)P (t | x)P (x)P (t | x)

x=

X P (y, t, x)
x=

X P (y, t, x)
x=

P (t | x)

X
x

P (t)

P (y, x | t)

= P (y | t)

Brady Neal

The magic of randomized experiments

10 / 40

Covariate balance implies association is causationd

Let X be a sufficient adjustment set

P (X | T = 1) = P (X | T = 0) =) T ?? X

P (y | do(t)) ==Xx

P (y | t, x)P (x)X P (y | t, x)P (t | x)P (x)P (t | x)

x=

X P (y, t, x)
x=

X P (y, t, x)
x=

P (t | x)

X
x

P (t)

P (y, x | t)

= P (y | t)

Brady Neal

The magic of randomized experiments

10 / 40

Covariate balance implies association is causationd

Let X be a sufficient adjustment set

P (X | T = 1) = P (X | T = 0) =) T ?? X

P (y | do(t)) ==Xx

P (y | t, x)P (x)X P (y | t, x)P (t | x)P (x)P (t | x)

x=

X P (y, t, x)
x=

X P (y, t, x)
x=

P (t | x)

X
x

P (t)

P (y, x | t)

= P (y | t)

Brady Neal

The magic of randomized experiments

10 / 40

Covariate balance implies association is causationd

Let X be a sufficient adjustment set

P (X | T = 1) = P (X | T = 0) =) T ?? X

P (y | do(t)) ==Xx

P (y | t, x)P (x)X P (y | t, x)P (t | x)P (x)P (t | x)

x=

X P (y, t, x)
x=

X P (y, t, x)
x=

P (t | x)

X
x

P (t)

P (y, x | t)

= P (y | t)

Brady Neal

The magic of randomized experiments

10 / 40

Covariate balance implies association is causationd

Let X be a sufficient adjustment set

P (X | T = 1) = P (X | T = 0) =) T ?? X

P (y | do(t)) ==Xx

P (y | t, x)P (x)X P (y | t, x)P (t | x)P (x)P (t | x)

x=

X P (y, t, x)
x=

X P (y, t, x)
x=

P (t | x)

X
x

P (t)

P (y, x | t)

= P (y | t)

Brady Neal

The magic of randomized experiments

10 / 40

Covariate balance implies association is causationd

Let X be a sufficient adjustment set

P (X | T = 1) = P (X | T = 0) =) T ?? X

P (y | do(t)) ==Xx

P (y | t, x)P (x)X P (y | t, x)P (t | x)P (x)P (t | x)

x=

X P (y, t, x)
x=

X P (y, t, x)
x=

P (t | x)

X
x

P (t)

P (y, x | t)

= P (y | t)

Brady Neal

The magic of randomized experiments

10 / 40

Covariate balance implies association is causationd

Let X be a sufficient adjustment set

P (X | T = 1) = P (X | T = 0) =) T ?? X

P (y | do(t)) ==Xx

P (y | t, x)P (x)X P (y | t, x)P (t | x)P (x)P (t | x)

x=

X P (y, t, x)
x=

X P (y, t, x)
x=

P (t | x)

X
x

P (t)

P (y, x | t)

= P (y | t)

Brady Neal

The magic of randomized experiments

10 / 40

Covariate balance implies association is causationd

Let X be a sufficient adjustment set

P (X | T = 1) = P (X | T = 0) =) T ?? X

P (y | do(t)) ==Xx

P (y | t, x)P (x)X P (y | t, x)P (t | x)P (x)P (t | x)

x=

X P (y, t, x)
x=

X P (y, t, x)
x=

P (t | x)

X
x

P (t)

P (y, x | t)

= P (y | t)

Brady Neal

The magic of randomized experiments

10 / 40

Covariate balance implies association is causationd

Let X be a sufficient adjustment set

P (X | T = 1) = P (X | T = 0) =) T ?? X

P (y | do(t)) ==Xx

P (y | t, x)P (x)X P (y | t, x)P (t | x)P (x)P (t | x)

x=

X P (y, t, x)
x=

X P (y, t, x)
x=

P (t | x)

X
x

P (t)

P (y, x | t)

= P (y | t)

Brady Neal

The magic of randomized experiments

10 / 40

Exchangeability

T =1

s
d
a
e
h

E[Y | T = 1] = y1
Brady Neal

tai
ls

T =0

E[Y | T = 0] = y0

The magic of randomized experiments

11 / 40

Exchangeability

T =1

s
l
i
ta

E[Y | T = 1] = y1
Brady Neal

he
ad

s

T =0

E[Y | T = 0] = y0

The magic of randomized experiments

11 / 40

Exchangeability

T =1

s
l
i
ta

E[Y | T = 1] = y1
Brady Neal

he
ad

s

T =0

E[Y | T = 0] = y0

The magic of randomized experiments

11 / 40

Question:
Write down the formal definition of(mean) exchangeability. Then, prove that
this yields “association is causation.”No backdoor paths

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T

Brady NealYThe magic of randomized experiments

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No backdoor paths
Confounding association
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Brady NealYThe magic of randomized experiments

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No backdoor paths
Confounding association
X

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Brady NealYThe magic of randomized experiments

13 / 40

No backdoor paths

X

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Brady NealYThe magic of randomized experiments

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Question:
What previous result tells us thatassociation is causation in this graph?
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Y

The magic of randomized experiments
Frontdoor adjustment
Pearl’s do-calculus
Determining identifiability from the graph

Brady Neal

Frontdoor adjustment

15 / 40

Recall the backdoor adjustment
P (Y | t)
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Causal association

Brady Neal

Frontdoor adjustment

16 / 40

Recall the backdoor adjustment
P (Y | t, c, w2 )
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Causal association

Brady Neal

Frontdoor adjustment

16 / 40

Recall the backdoor adjustment
P (Y | t, c, w2 )W2<latexit sha1_base64="LQXcL0mQbxU7nAhuYYqjJ+F5HZQ=">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</latexit>

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W3

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Y

Causal association

Brady Neal

Frontdoor adjustment

16 / 40

Frontdoor adjustment: big picture<latexit sha1_base64="rmzf2dnvrtQDlEmd/bgMDday4FU=">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</latexit>

confounding association
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Y

causal association

Brady Neal

Frontdoor adjustment

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Frontdoor adjustment: big picture<latexit sha1_base64="+9t/9tyzRe4QnrnlIJ4LymjoViE=">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</latexit>

W
focus
T

M

Y

only causal association

Brady Neal

Frontdoor adjustment

17 / 40

Frontdoor adjustment: big picture<latexit sha1_base64="997OV9zRW5X1PA5ILGqON3gwUec=">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</latexit>

W

T

Brady Neal

Frontdoor adjustment

MY

17 / 40

Frontdoor adjustment: big picture
1. Identify the causal effect of T on M<latexit sha1_base64="jaG7k2a3PUqW/u74qXTDqaNxUVs=">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</latexit>

W

T

Brady Neal

Step 1

Frontdoor adjustment

MY

17 / 40

Frontdoor adjustment: big picture
1. Identify the causal effect of T on M
2. Identify the causal effect of M on Y<latexit sha1_base64="s/lzlHOyX5PtbL+vjpSExh86iHY=">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</latexit>

W

T

Brady Neal

Step 1

Frontdoor adjustment

M

Step 2Y

17 / 40

Frontdoor adjustment: big picture
1. Identify the causal effect of T on M
2. Identify the causal effect of M on Y
3. Combine the above steps to identify
the causal effect of T on Y<latexit sha1_base64="W2GtS6RWfVw5rxfaIh3GklPLAi0=">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</latexit>

W

T

Step 1

M

Step 2

Y

Step 3

Brady Neal

Frontdoor adjustment

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Frontdoor adjustment: step 1
Identify the causal effect of T on M<latexit sha1_base64="jaG7k2a3PUqW/u74qXTDqaNxUVs=">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</latexit>

W

T

Step 1

Brady Neal

M

Y

Frontdoor adjustment

18 / 40

Frontdoor adjustment: step 1
Identify the causal effect of T on M
P (m | do(t)) = P (m | t)<latexit sha1_base64="G8X2gIMu+wYddwsRzOv8TGJFb5Q=">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</latexit><latexit sha1_base64="jaG7k2a3PUqW/u74qXTDqaNxUVs=">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</latexit>

W

T

Step 1

Brady Neal

M

Y

Frontdoor adjustment

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Frontdoor adjustment: step 1
Identify the causal effect of T on M
P (m | do(t)) = P (m | t)<latexit 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sha1_base64="jaG7k2a3PUqW/u74qXTDqaNxUVs=">AAAVznicrVhLb9tGEN7EfaTuK07hUy9sbAMJQKuSHSBBAgEBnARF0bQJ6sRJLSOgyJVEiK+QqyiOIPTav9Ef0nOv7bH/pjPfLkVSbweRIHI5O/PNY2eGu2ongZ+pev2/S5c3Pvr4k0+vfLb5+RdffvX11a1rz7N4kLrymRsHcfqi7WQy8CP5TPkqkC+SVDphO5An7f4Rz5+8kWnmx9GxOk/kWeh0I7/ju44i0qutjbuttuz60Uj5/XeJ76pBKseblvnsWT/Hnswmz62IHk8z5Shpe07Wk96ZdaPlxlEnHkQeqblpOcq6UbfrN63Rbnlid3xvHgqLh9LzHRWnN63TtgziodWMO1ZZ9oyxcq4lQIocV6GMFCEFsqM0UC4IlAnLEph4oNw4lASS+t3ePBTDkWOUwvXQ65bDlTiqNx0gSSwVY++txZ9bNctdchu88AaRJGN/VTKxGuNymAuEvRyiWIIFCAfjuSbsLbJhrroF2uYBl0MaBH6SyZJOmNfxVbOstwRbANqW1NK25aXO0LbeZq4TyGa9dse2zvW4UWvYVuCQu024fBcOH1ImZDQ4pEQ2y9ySkVcpk1dXd+q1Oj7W7KBhBjvCfJ7EW1f+Ei3hiVi4YiBCIUUkFI0D4YiMvqeiIeoiIdqZGBEtpZGPeSnGYpNkB8QlicMhap+uXXo6NdSInhkzg7RLWgL6pSRpiT3D49G4A6q+s36rxLtIxwjYbOM53dsGMySqEj2irpLLOdeVY58UWXgHvvhkZwIKe+lWPOrQPaBnRfbz9Zw4JY08kkpp5BItIKqmsI6U7jqu7HkPcXbAJ2nENi32pk22L14Jno/p2ycsh8YZLGVbLfHIRD2CZglbmSfAii1GfEseavuWYXUmPvhYVe0F8x4TpS/e0aiKvExnOXcWcymDPJ6ji/kVOCLizogSI699ip9PHNVMdAmTdThYyS58SZAfNYP8I+bzXE1oPfdhT4ZVs5BNPvQkpm4Ku3M7A7q3gZ2S/IjmetDDsbDJHgnsFLmZR9EGd0pPQzy5sNGdotdQd7yeNvnE2WUTbkwz/gRLR0LHM7dIWzcimmW+++In6Je07jaiwnltk2wMC3UWZ9AVmbVvUtfguIR0ZSpHNqfY0BHCjh6ypU+0Qp9GYn3fkxcZ6icwqCO6B4haAhQb2jk+Q4xD+MQrOoDuIY096GFpizpZTdw2dowrWjk2PnrPrFYL+ce1q/vetFaWi1CplsnKJjTVxS3jMa8A212Ommt6ne64GbwbT1nVhv1F1+AcrdrHUsX6VNfKQzb2IFXF5SjO93OeLweoExtrybHvEkcTUh1Dz1CbidG0Waq/I1MlMdbDxT00OaHrnbUNp2ZGRA9NpALi03XObxyeeU2YDnKqheiV55chPjTyea9v03cE6nK556isqmxKo9FkZrm8j5FEfPSIV1zjJYh1Al++m/ys0sy62B8GL5pB5IxVBmlddA+5Md/zY7FDvWCHsqoaT15ppqTYeyQ03p1w7hIu8/ahJSIKV/hoMj9euQYPiDI2EetgLq/DwgYfnc8DZ8tknubQFaWmOLiql2vlvJyXN5q+XNab5HtVVtOXyzLlDXLTN/0gA9aLFXIK68ASYSlLj1dIxXgnaLq2+eUa9kV41w5MXqTGwpMVkiF6jPYqNlX5eKV9Cl0wNTFhmd/eI4Jv0P/G2AFdNI6FrLpQNAu58/eKaSE/vGBkC8nwgvEtJN+tkJR4N/gTGss8XVlTzPVkJfLryb5EV5GNa5uiw+9iz3Qcy7xHeLX1W6+J2rbNOyx/i+W7z6zU33xoL3amI+wihqYT5vuJfWjSvv1AtfSY3tE8PhKHtD9o0PVRpWuui8r79IGptzLuASHfpn73AH3x4riJ2U3vV/bcVR0PSUP+G5vTScEdmD3kIqSL6J2PrHPdm9nLTmvSu5Uecekd/PkKxKJ65uMtX9/5yPocHWMHWezjPwRy+YyyHl6RM4tPlKvOuYtPXwndO3QN0BEyw/sAWRegM0jxzJwQ9dv3aOasy/l7QFXBuTu6fm1KXxV/+vTUxelnQBwFj94FK0hrWhlx0TksgYwyu40MZ5biZFarnLQ03rRtWSmvQoPXNN1Kmq60+erqTmP6v5nZwclBrXGr1mg8Pdi5f8f8cXNFfCuuixsUqNviPpXmEwqsu/Hnxt8b/2z8u/3L9mB7vP27Zr18ych8Iyqf7T/+B3xymB8=</latexit>

W

T

Step 1

Brady Neal

M

Y

Frontdoor adjustment

18 / 40

Frontdoor adjustment: step 2
Identify the causal effect of M on Y<latexit sha1_base64="4qCWONnkrp2T9BdIkwGTnidGsek=">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</latexit>

W

T

Brady Neal

M

Step 2

Frontdoor adjustment

Y

19 / 40

Frontdoor adjustment: step 2
Identify the causal effect of M on Y<latexit sha1_base64="QG57jZbv7M0gSwvR6kQ0evEFBxI=">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</latexit>

P (y | do(m)) =Xt

P (y | m, t) P (t)<latexit sha1_base64="4qCWONnkrp2T9BdIkwGTnidGsek=">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</latexit>

W

T

Brady Neal

M

Step 2

Frontdoor adjustment

Y

19 / 40

Frontdoor adjustment: step 2
Identify the causal effect of M on Y<latexit sha1_base64="QG57jZbv7M0gSwvR6kQ0evEFBxI=">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</latexit>

P (y | do(m)) =Xt

P (y | m, t) P (t)<latexit sha1_base64="etULFHFN9sWdeuOCtSaayGN9DhY=">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</latexit>

W

T

Brady Neal

M

Step 2

Frontdoor adjustment

Y

19 / 40

Frontdoor adjustment: step 2
Identify the causal effect of M on Y<latexit sha1_base64="QG57jZbv7M0gSwvR6kQ0evEFBxI=">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</latexit><latexit sha1_base64="QG57jZbv7M0gSwvR6kQ0evEFBxI=">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</latexit>

P (y | do(m)) =Xt

P (y | m, t) P (t)<latexit sha1_base64="etULFHFN9sWdeuOCtSaayGN9DhY=">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</latexit>

W

T

Brady Neal

M

Step 2

Frontdoor adjustment

Y

19 / 40

Frontdoor adjustment: step 3
Combine steps 1 and 2 to identify the causal effect of T on Y<latexit sha1_base64="2uDt9fKu4fa/U8cv7ZLSawAUzd4=">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</latexit>

W

T

M

Y

Step 3
Brady Neal

Frontdoor adjustment

20 / 40

Frontdoor adjustment: step 3
Combine steps 1 and 2 to identify the causal effect of T on Y
Goal<latexit sha1_base64="TimGMbSX9Ug74mo/kChlzWTeYlU=">AAATIXicrVjbbttGEF2nN9e9Oepb+8JGLhADsiq5BpwXAQHsBEXRAA5gx26jwJAoyiLEW0jKriPoF/oR/Y++97HoW9Hf6Af0zNmVSMq6BpEgcjk7c2Z2btxVO/LcJK3V/tm49977H3z40ebHW598+tnnX2zfL71IwkFsO2d26IXxRbuVOJ4bOGepm3rORRQ7Lb/tOeft/pHMn187ceKGwWl6Gzmv/NZV4HZdu5WCdLn9WzN62AwHqR36znXLs5q+27Gax+HDZgqY1HeCFOTdXathNZOBf9n0nY7bSsOYzJAtPM8TblasuXpyALu7l9vlWrXGj3V3UDeDsjKfk/D+5h+qqToqVLYaKF85KlApxp5qqQTfl6quaioC7ZUaghZj5HLeUSO1BdkBuBxwtEDt43qFp5eGGuBZMBNK29Di4RdD0lLfGp4Oxl1S9V30WzneeTqGxBYbb3FvG0wf1FT1QF0mN+ZcVU7WlMLCR1yLCzsjUmSVdmFFXdw9PKewX6634HQw6kAqxsgGzQNVU0RHjLv2q6y8Rz+3yOdgJDbNX00bts+PhMyH+PaB1cI4oaViq6WeGq8H1OzQVuHxGLH5iL9ihdq+RVjdyRpcRlWvQnhPQemrNxgVkRfpzOfOfK7UII9m6BL+lBwBuBNQQua1C/+54Chmog1M0dFiJK+4loj5UTXIP3J+nKsR4rlHexJGzWI2udQTmbrJ7B7b6eHeJnYM+SHmetQjvqjAHofYMXNz7MUKuWM83fDJpo32FL3KupN4VrAmya4KcEPMuBMs7Qntz7FF2rohaJb57qmfqN9B3Cv0iuR1BbIhLdRZnFBXYGLfQNcQv/i4ClU8O6ZUqMOnHT1mSx+0TJ9GEn3fYRUJ68czqEPcPXotIkqF2sU/Nxz7XJNEdEDdNxh3qEekLXSyqjo0dowKWsU3LnvPXa0W809qV/e9aa0iF7BSLZOVDWqqqQOzYomA2J33mm16ne64CVc3mrKqTfuzriE5WrRPpLL4FGPVYTb2KFXEFS/OXuesteyzTiqMpfj+ChwNSnUNPWFtRkbTVq7+jkyVhIyHzbtvckLXu2i7mZoZgu4bT3ng03UubxyZeQ3MFnOqSe/l5xchPjHy417fxndI6mK5F6ysomyM0XAys1je5cihf/RIIq7xIvo64lq+mfys3Myq2O8GL7iDKBmbGqRV0TvMjdkrP1Vl9IIysqroT4m0UGLuPSKMdyacO8AV3j61BKBIhQ8n86OlMTgGZWQ81uXcuA4zG1x2vg45mybzNIeuqHSKQ6p6sVbJy1l5o+mLZTuTfC/KavpiWaFcMzdd0w8SYl0skUsZB5Hwc1l6ukQq5DtB07XNP69gX8B37cDkRWwsPF8i6bPH6FWFpiqfLbUvZReMjU9E5pe38OA1+9+IO6B1/ZjJpmt5M5O7fSufZvI3a3o2k/TX9G8m+WaJpMN3gzuhiczzpTUlXCdLkV9P9iW6iiq8tuEdeRd3TMexzHtEoq3feg3WdsW8w8ZvsfHuM8n1N5fas53pkLuIG9MJx/uJPWrSa/sBtfQM72gZH6nvsT+o4/q00DVXRZV9+sDUWx53H8iH6HfH7Ivr40ZmN71X2HMXdTyBhvFvZE4nGbdn9pDzkNbROxtZ53rnzl52WpPerfTApXfwt0sQs+qZjbc4vrOR9Tk65A4y28e/C+T8GWU1vCxn5p8ol51z55++Ity7uHrsCInhPWbWeewMjjozJ0T99j26c9aV/N1HVUjuDh+UpvQV8adPT1c8/QzAkfHoXXBKaU3LI847h0WUSc1uI+GZJTuZVQsnLY03bVuSyyvf4DVMt3JMV9q63C7Xp/+buTs436/WD6r1+vOD8uNH5o+bTfW1eqAewlGH6jFK8wSOtdV/G19tlDd2Sr+X/iz9Vfpbs97bMDJfqsKn9O//HlTHOw==</latexit>

P (y | do(t)) =Xm

P (m | do(t)) P (y | do(m))<latexit sha1_base64="2uDt9fKu4fa/U8cv7ZLSawAUzd4=">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</latexit>

W

T

M

Y

Step 3
Brady Neal

Frontdoor adjustment

20 / 40

Frontdoor adjustment: step 3
Combine steps 1 and 2 to identify the causal effect of T on Y
Goal<latexit sha1_base64="TimGMbSX9Ug74mo/kChlzWTeYlU=">AAATIXicrVjbbttGEF2nN9e9Oepb+8JGLhADsiq5BpwXAQHsBEXRAA5gx26jwJAoyiLEW0jKriPoF/oR/Y++97HoW9Hf6Af0zNmVSMq6BpEgcjk7c2Z2btxVO/LcJK3V/tm49977H3z40ebHW598+tnnX2zfL71IwkFsO2d26IXxRbuVOJ4bOGepm3rORRQ7Lb/tOeft/pHMn187ceKGwWl6Gzmv/NZV4HZdu5WCdLn9WzN62AwHqR36znXLs5q+27Gax+HDZgqY1HeCFOTdXathNZOBf9n0nY7bSsOYzJAtPM8TblasuXpyALu7l9vlWrXGj3V3UDeDsjKfk/D+5h+qqToqVLYaKF85KlApxp5qqQTfl6quaioC7ZUaghZj5HLeUSO1BdkBuBxwtEDt43qFp5eGGuBZMBNK29Di4RdD0lLfGp4Oxl1S9V30WzneeTqGxBYbb3FvG0wf1FT1QF0mN+ZcVU7WlMLCR1yLCzsjUmSVdmFFXdw9PKewX6634HQw6kAqxsgGzQNVU0RHjLv2q6y8Rz+3yOdgJDbNX00bts+PhMyH+PaB1cI4oaViq6WeGq8H1OzQVuHxGLH5iL9ihdq+RVjdyRpcRlWvQnhPQemrNxgVkRfpzOfOfK7UII9m6BL+lBwBuBNQQua1C/+54Chmog1M0dFiJK+4loj5UTXIP3J+nKsR4rlHexJGzWI2udQTmbrJ7B7b6eHeJnYM+SHmetQjvqjAHofYMXNz7MUKuWM83fDJpo32FL3KupN4VrAmya4KcEPMuBMs7Qntz7FF2rohaJb57qmfqN9B3Cv0iuR1BbIhLdRZnFBXYGLfQNcQv/i4ClU8O6ZUqMOnHT1mSx+0TJ9GEn3fYRUJ68czqEPcPXotIkqF2sU/Nxz7XJNEdEDdNxh3qEekLXSyqjo0dowKWsU3LnvPXa0W809qV/e9aa0iF7BSLZOVDWqqqQOzYomA2J33mm16ne64CVc3mrKqTfuzriE5WrRPpLL4FGPVYTb2KFXEFS/OXuesteyzTiqMpfj+ChwNSnUNPWFtRkbTVq7+jkyVhIyHzbtvckLXu2i7mZoZgu4bT3ng03UubxyZeQ3MFnOqSe/l5xchPjHy417fxndI6mK5F6ysomyM0XAys1je5cihf/RIIq7xIvo64lq+mfys3Myq2O8GL7iDKBmbGqRV0TvMjdkrP1Vl9IIysqroT4m0UGLuPSKMdyacO8AV3j61BKBIhQ8n86OlMTgGZWQ81uXcuA4zG1x2vg45mybzNIeuqHSKQ6p6sVbJy1l5o+mLZTuTfC/KavpiWaFcMzdd0w8SYl0skUsZB5Hwc1l6ukQq5DtB07XNP69gX8B37cDkRWwsPF8i6bPH6FWFpiqfLbUvZReMjU9E5pe38OA1+9+IO6B1/ZjJpmt5M5O7fSufZvI3a3o2k/TX9G8m+WaJpMN3gzuhiczzpTUlXCdLkV9P9iW6iiq8tuEdeRd3TMexzHtEoq3feg3WdsW8w8ZvsfHuM8n1N5fas53pkLuIG9MJx/uJPWrSa/sBtfQM72gZH6nvsT+o4/q00DVXRZV9+sDUWx53H8iH6HfH7Ivr40ZmN71X2HMXdTyBhvFvZE4nGbdn9pDzkNbROxtZ53rnzl52WpPerfTApXfwt0sQs+qZjbc4vrOR9Tk65A4y28e/C+T8GWU1vCxn5p8ol51z55++Ity7uHrsCInhPWbWeewMjjozJ0T99j26c9aV/N1HVUjuDh+UpvQV8adPT1c8/QzAkfHoXXBKaU3LI847h0WUSc1uI+GZJTuZVQsnLY03bVuSyyvf4DVMt3JMV9q63C7Xp/+buTs436/WD6r1+vOD8uNH5o+bTfW1eqAewlGH6jFK8wSOtdV/G19tlDd2Sr+X/iz9Vfpbs97bMDJfqsKn9O//HlTHOw==</latexit><latexit sha1_base64="TimGMbSX9Ug74mo/kChlzWTeYlU=">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</latexit>

P (y | do(t)) =Xm

P (m | do(t)) P (y | do(m))<latexit sha1_base64="xDdwfY1IrXBnqW4i4OAHPMiIuwo=">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</latexit>

W

T

Step 1

M

Y

Step 3
Brady Neal

Frontdoor adjustment

20 / 40

Frontdoor adjustment: step 3
Combine steps 1 and 2 to identify the causal effect of T on Y
Goal<latexit sha1_base64="TimGMbSX9Ug74mo/kChlzWTeYlU=">AAATIXicrVjbbttGEF2nN9e9Oepb+8JGLhADsiq5BpwXAQHsBEXRAA5gx26jwJAoyiLEW0jKriPoF/oR/Y++97HoW9Hf6Af0zNmVSMq6BpEgcjk7c2Z2btxVO/LcJK3V/tm49977H3z40ebHW598+tnnX2zfL71IwkFsO2d26IXxRbuVOJ4bOGepm3rORRQ7Lb/tOeft/pHMn187ceKGwWl6Gzmv/NZV4HZdu5WCdLn9WzN62AwHqR36znXLs5q+27Gax+HDZgqY1HeCFOTdXathNZOBf9n0nY7bSsOYzJAtPM8TblasuXpyALu7l9vlWrXGj3V3UDeDsjKfk/D+5h+qqToqVLYaKF85KlApxp5qqQTfl6quaioC7ZUaghZj5HLeUSO1BdkBuBxwtEDt43qFp5eGGuBZMBNK29Di4RdD0lLfGp4Oxl1S9V30WzneeTqGxBYbb3FvG0wf1FT1QF0mN+ZcVU7WlMLCR1yLCzsjUmSVdmFFXdw9PKewX6634HQw6kAqxsgGzQNVU0RHjLv2q6y8Rz+3yOdgJDbNX00bts+PhMyH+PaB1cI4oaViq6WeGq8H1OzQVuHxGLH5iL9ihdq+RVjdyRpcRlWvQnhPQemrNxgVkRfpzOfOfK7UII9m6BL+lBwBuBNQQua1C/+54Chmog1M0dFiJK+4loj5UTXIP3J+nKsR4rlHexJGzWI2udQTmbrJ7B7b6eHeJnYM+SHmetQjvqjAHofYMXNz7MUKuWM83fDJpo32FL3KupN4VrAmya4KcEPMuBMs7Qntz7FF2rohaJb57qmfqN9B3Cv0iuR1BbIhLdRZnFBXYGLfQNcQv/i4ClU8O6ZUqMOnHT1mSx+0TJ9GEn3fYRUJ68czqEPcPXotIkqF2sU/Nxz7XJNEdEDdNxh3qEekLXSyqjo0dowKWsU3LnvPXa0W809qV/e9aa0iF7BSLZOVDWqqqQOzYomA2J33mm16ne64CVc3mrKqTfuzriE5WrRPpLL4FGPVYTb2KFXEFS/OXuesteyzTiqMpfj+ChwNSnUNPWFtRkbTVq7+jkyVhIyHzbtvckLXu2i7mZoZgu4bT3ng03UubxyZeQ3MFnOqSe/l5xchPjHy417fxndI6mK5F6ysomyM0XAys1je5cihf/RIIq7xIvo64lq+mfys3Myq2O8GL7iDKBmbGqRV0TvMjdkrP1Vl9IIysqroT4m0UGLuPSKMdyacO8AV3j61BKBIhQ8n86OlMTgGZWQ81uXcuA4zG1x2vg45mybzNIeuqHSKQ6p6sVbJy1l5o+mLZTuTfC/KavpiWaFcMzdd0w8SYl0skUsZB5Hwc1l6ukQq5DtB07XNP69gX8B37cDkRWwsPF8i6bPH6FWFpiqfLbUvZReMjU9E5pe38OA1+9+IO6B1/ZjJpmt5M5O7fSufZvI3a3o2k/TX9G8m+WaJpMN3gzuhiczzpTUlXCdLkV9P9iW6iiq8tuEdeRd3TMexzHtEoq3feg3WdsW8w8ZvsfHuM8n1N5fas53pkLuIG9MJx/uJPWrSa/sBtfQM72gZH6nvsT+o4/q00DVXRZV9+sDUWx53H8iH6HfH7Ivr40ZmN71X2HMXdTyBhvFvZE4nGbdn9pDzkNbROxtZ53rnzl52WpPerfTApXfwt0sQs+qZjbc4vrOR9Tk65A4y28e/C+T8GWU1vCxn5p8ol51z55++Ity7uHrsCInhPWbWeewMjjozJ0T99j26c9aV/N1HVUjuDh+UpvQV8adPT1c8/QzAkfHoXXBKaU3LI847h0WUSc1uI+GZJTuZVQsnLY03bVuSyyvf4DVMt3JMV9q63C7Xp/+buTs436/WD6r1+vOD8uNH5o+bTfW1eqAewlGH6jFK8wSOtdV/G19tlDd2Sr+X/iz9Vfpbs97bMDJfqsKn9O//HlTHOw==</latexit><latexit 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P (y | do(t)) =Xm

P (m | do(t)) P (y | do(m))<latexit sha1_base64="W2GtS6RWfVw5rxfaIh3GklPLAi0=">AAAVzXicrVhbj9NGFB7YXuj2xlLtU19cdlcCyZsmu0jQokhIC6iqCgJ1YWk3K+TYk8SKb9gTwpKmr/0b/SN97Wv72n/Tc74Zx3bui0gUe3zmnO/cxzNpJ4GfqXr9v0uXNz748KOPr3yy+elnn3/x5dWta8+zeJC68pkbB3H6ou1kMvAj+Uz5KpAvklQ6YTuQJ+3+Ec+fvJZp5sfRsTpP5FnodCO/47uOItLLrY3vWm3Z9aOR8vtvE99Vg1SONy3z2bMex57MJs+tiB5PM+UoaXtO1pPemXWj5cZRJx5EHqm5aTnKulG36zet0W55Ynd8dx4Ki4fS8x0Vpzet07YM4qHVjDtWWfaMsXKuJUCKHFehjBQhBbKjNFAuCJQJyxKYeKDcOJQEkvrd3jwUw5FjlML1wOuWw5U4qjcdIEksFWPvrsWfWzXLXXIbvPAGkSRjf1YysRrjcphnEYoMLAA4GM+1YG+RCXO17S1QNw+5HNIg8JNMTmWr46tmWW8JtcCzLamFbctLnaFtvclcJ5DNeu2ObZ3rcaPWsK3AIXebcPl7OHxIhZDR4JDq2GS5JSOv0iUvr+7Ua3V8rNlBwwx2hPk8ibeu/CVawhOxcMVAhEKKSCgaB8IRGX1PRUPURUK0MzEiWkojH/NSjMUmyQ6ISxKHQ9Q+Xbv0dGqoET0zZgZpl7QE9EtJ0hJ7hsejcQdUfWf9Vol3kY4RsNnGc7q3DWZIVCV6RF0ll3OuK8c+KbLwDnzxyc4EFPbSrXjUoXtAz4rs5+s5cUoaeSSV0sglWkBUTWEdKd11XNnzHuLsgE/SiG1a7E2bbF+cCZ6P6dsnLIfGGSxlWy3x0EQ9gmYJW5knQMYWI74hD7V9y7A6Ex98ZFV7wbzHROmLtzSqIi/TWa6dxVzKII/n6GJ+BY6IuDOixKhrn+LnE0e1El3CZB0OMtmFLwnqo2aQf8R8XqsJ5XMf9mTImoVq8qEnMX1T2J3bGdC9DeyU5Ec014MejoVN9khgp6jNPIo2uFN6GuLJhY3uFL2GvuN82uQTV5dNuDHN+BMsHQkdz9wibd2IaJb57oufoF9S3m1EhevaJtkYFuoqzqArMrlv0qrBcQnpylSObE6xoSOEHT1US59ohT6NxPq+JS8y9E9gUEd0DxC1BCg2tHN8hhiH8IkzOoDuIY096GFpi1aymrht7BhXtHJsfKw9s1ot1B/3rl73prWyXIROtUxVNqGpLm4ZjzkDbHc5aq5Z6/SKm8G78ZRVbdhfrBpco1X7WKrITzVXHqqxB6kqLkdxvp/zfDlAn9jIJce+SxxNSHUMPUNvJkbTZqn/jkyXxMiHi3toakL3O2sbTs2MiB6aSAXEp/uc3zg884owHdRUC9Erzy9DfGDk87W+Td8RqMvlnqOzqrIpjUaTmeXyPkYS8dEjzrjGSxDrBL58M/lZpZl1sd8PXjSDyBWrDNK66B5qY77nx2KH1oIdqqpqPDnTTEmx90hovDvh3CVc5u1DS0QU7vDRZH68Mgf3iTI2EetgLu/DwgYfK58HzpapPM2hO0pNcXBXL9fKdTmvbjR9uaw3qfeqrKYvl2XKa9Smb9aDDFgvVsgp5IElwlKVHq+QivFO0HRt8y9r2BfhXTswdZEaC09WSIZYY7RXsenKRyvtU1gFUxMTlvn1HSL4GuvfGDugi8axkFUXimYhd/5OMS3khxeMbCEZXjC+heTbFZIS7wZ/QmOZpyt7irmerER+NdmX6C6ycW1TdPhd7JkVxzLvEc62fus10du2eYflb7F895mV1jcf2oud6Qi7iKFZCfP9xD40ad9+oF56RO9oHh+JQ9ofNOj6sLJqrovK+/SB6bcy7gEh36b17j7WxYvjJmY3vV/Zc1d1PCAN+W9sTicFd2D2kIuQLqJ3PrKudW9mLzutSe9WesSld/DnKxCL7pmPtzy/85H1OTrGDrLYx78P5PIZZT28omYWnyhXnXMXn74SunfoGmBFyAzvfVRdgJVBimfmhKjfvkczZ12u3wPqCq7d0fVrU/qq+NOnpy5OPwPiKHj0LlhBWtPKiIvOYQlklNltZDizFCezWuWkpfGmbctKdRUavKZZraRZlTZfXt1pTP83Mzs4Oag1btUajacHO/fumD9uroivxXVxgwJ1W9yj1nxCgXU3/tz4e+OfjX+3H2+r7d+2f9esly8Zma9E5bP9x//MG5fw</latexit>

W

T

Step 1

M

Step 2

Y

Step 3
Brady Neal

Frontdoor adjustment

20 / 40

Frontdoor adjustment: step 3
Combine steps 1 and 2 to identify the causal effect of T on Y
Goal<latexit 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P (y | do(t)) =Xm

P (m | do(t)) P (y | do(m))<latexit sha1_base64="W2GtS6RWfVw5rxfaIh3GklPLAi0=">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</latexit>

W

T

Step 1

M

Step 2

Y

Step 3
Brady Neal

Frontdoor adjustment

20 / 40

Frontdoor adjustment: step 3
Combine steps 1 and 2 to identify the causal effect of T on Y
Goal<latexit sha1_base64="TimGMbSX9Ug74mo/kChlzWTeYlU=">AAATIXicrVjbbttGEF2nN9e9Oepb+8JGLhADsiq5BpwXAQHsBEXRAA5gx26jwJAoyiLEW0jKriPoF/oR/Y++97HoW9Hf6Af0zNmVSMq6BpEgcjk7c2Z2btxVO/LcJK3V/tm49977H3z40ebHW598+tnnX2zfL71IwkFsO2d26IXxRbuVOJ4bOGepm3rORRQ7Lb/tOeft/pHMn187ceKGwWl6Gzmv/NZV4HZdu5WCdLn9WzN62AwHqR36znXLs5q+27Gax+HDZgqY1HeCFOTdXathNZOBf9n0nY7bSsOYzJAtPM8TblasuXpyALu7l9vlWrXGj3V3UDeDsjKfk/D+5h+qqToqVLYaKF85KlApxp5qqQTfl6quaioC7ZUaghZj5HLeUSO1BdkBuBxwtEDt43qFp5eGGuBZMBNK29Di4RdD0lLfGp4Oxl1S9V30WzneeTqGxBYbb3FvG0wf1FT1QF0mN+ZcVU7WlMLCR1yLCzsjUmSVdmFFXdw9PKewX6634HQw6kAqxsgGzQNVU0RHjLv2q6y8Rz+3yOdgJDbNX00bts+PhMyH+PaB1cI4oaViq6WeGq8H1OzQVuHxGLH5iL9ihdq+RVjdyRpcRlWvQnhPQemrNxgVkRfpzOfOfK7UII9m6BL+lBwBuBNQQua1C/+54Chmog1M0dFiJK+4loj5UTXIP3J+nKsR4rlHexJGzWI2udQTmbrJ7B7b6eHeJnYM+SHmetQjvqjAHofYMXNz7MUKuWM83fDJpo32FL3KupN4VrAmya4KcEPMuBMs7Qntz7FF2rohaJb57qmfqN9B3Cv0iuR1BbIhLdRZnFBXYGLfQNcQv/i4ClU8O6ZUqMOnHT1mSx+0TJ9GEn3fYRUJ68czqEPcPXotIkqF2sU/Nxz7XJNEdEDdNxh3qEekLXSyqjo0dowKWsU3LnvPXa0W809qV/e9aa0iF7BSLZOVDWqqqQOzYomA2J33mm16ne64CVc3mrKqTfuzriE5WrRPpLL4FGPVYTb2KFXEFS/OXuesteyzTiqMpfj+ChwNSnUNPWFtRkbTVq7+jkyVhIyHzbtvckLXu2i7mZoZgu4bT3ng03UubxyZeQ3MFnOqSe/l5xchPjHy417fxndI6mK5F6ysomyM0XAys1je5cihf/RIIq7xIvo64lq+mfys3Myq2O8GL7iDKBmbGqRV0TvMjdkrP1Vl9IIysqroT4m0UGLuPSKMdyacO8AV3j61BKBIhQ8n86OlMTgGZWQ81uXcuA4zG1x2vg45mybzNIeuqHSKQ6p6sVbJy1l5o+mLZTuTfC/KavpiWaFcMzdd0w8SYl0skUsZB5Hwc1l6ukQq5DtB07XNP69gX8B37cDkRWwsPF8i6bPH6FWFpiqfLbUvZReMjU9E5pe38OA1+9+IO6B1/ZjJpmt5M5O7fSufZvI3a3o2k/TX9G8m+WaJpMN3gzuhiczzpTUlXCdLkV9P9iW6iiq8tuEdeRd3TMexzHtEoq3feg3WdsW8w8ZvsfHuM8n1N5fas53pkLuIG9MJx/uJPWrSa/sBtfQM72gZH6nvsT+o4/q00DVXRZV9+sDUWx53H8iH6HfH7Ivr40ZmN71X2HMXdTyBhvFvZE4nGbdn9pDzkNbROxtZ53rnzl52WpPerfTApXfwt0sQs+qZjbc4vrOR9Tk65A4y28e/C+T8GWU1vCxn5p8ol51z55++Ity7uHrsCInhPWbWeewMjjozJ0T99j26c9aV/N1HVUjuDh+UpvQV8adPT1c8/QzAkfHoXXBKaU3LI847h0WUSc1uI+GZJTuZVQsnLY03bVuSyyvf4DVMt3JMV9q63C7Xp/+buTs436/WD6r1+vOD8uNH5o+bTfW1eqAewlGH6jFK8wSOtdV/G19tlDd2Sr+X/iz9Vfpbs97bMDJfqsKn9O//HlTHOw==</latexit><latexit sha1_base64="TimGMbSX9Ug74mo/kChlzWTeYlU=">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</latexit><latexit sha1_base64="TimGMbSX9Ug74mo/kChlzWTeYlU=">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</latexit><latexit sha1_base64="TimGMbSX9Ug74mo/kChlzWTeYlU=">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</latexit>

P (y | do(t)) =<latexit sha1_base64="yxpHQq/uB9mZMZxJMB22fH21j40=">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</latexit>

=

X
m

X
m

P (m | do(t)) P (y | do(m))
P (m | t)Xt0<latexit sha1_base64="W2GtS6RWfVw5rxfaIh3GklPLAi0=">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</latexit>

P (y | m, t0 ) P (t0 )T

W

Step 1

M

Step 2

Y

Step 3
Brady Neal

Frontdoor adjustment

20 / 40

The frontdoor adjustment and criterion<latexit sha1_base64="2tkLk9Yiao3kHPCdzl9N1f7aGpo=">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</latexit>

P (y | do(t)) =Brady NealXm

P (m | t)Xt0

P (y | m, t0 ) P (t0 )Frontdoor adjustment

21 / 40

The frontdoor adjustment and criterion
If (T, M, Y) satisfy the frontdoor criterion, and we have positivity, then<latexit sha1_base64="2tkLk9Yiao3kHPCdzl9N1f7aGpo=">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</latexit>

P (y | do(t)) =Brady NealXm

P (m | t)Xt0

P (y | m, t0 ) P (t0 )Frontdoor adjustment

21 / 40

The frontdoor adjustment and criterion
If (T, M, Y) satisfy the frontdoor criterion, and we have positivity, then<latexit sha1_base64="2tkLk9Yiao3kHPCdzl9N1f7aGpo=">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</latexit>

P (y | do(t)) =Xm

P (m | t)Xt0

P (y | m, t0 ) P (t0 )A set of variables M satisfies the frontdoor criterion relative to T and Y
if the following are true:

Brady Neal

Frontdoor adjustment

21 / 40

The frontdoor adjustment and criterion
If (T, M, Y) satisfy the frontdoor criterion, and we have positivity, then<latexit sha1_base64="2tkLk9Yiao3kHPCdzl9N1f7aGpo=">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</latexit>

P (y | do(t)) =Xm

P (m | t)Xt0

P (y | m, t0 ) P (t0 )A set of variables M satisfies the frontdoor criterion relative to T and Y
if the following are true:
1. M completely mediates the effect of T on Y (i.e. all causal paths fromT to Y go through M).

Brady Neal

Frontdoor adjustment

21 / 40

The frontdoor adjustment and criterion
If (T, M, Y) satisfy the frontdoor criterion, and we have positivity, then<latexit sha1_base64="2tkLk9Yiao3kHPCdzl9N1f7aGpo=">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</latexit>

P (y | do(t)) =Xm

P (m | t)Xt0

P (y | m, t0 ) P (t0 )A set of variables M satisfies the frontdoor criterion relative to T and Y
if the following are true:
1. M completely mediates the effect of T on Y (i.e. all causal paths fromT to Y go through M).
2. There is no unblocked backdoor path from T to M.

Brady Neal

Frontdoor adjustment

21 / 40

The frontdoor adjustment and criterion
If (T, M, Y) satisfy the frontdoor criterion, and we have positivity, then<latexit sha1_base64="2tkLk9Yiao3kHPCdzl9N1f7aGpo=">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</latexit>

P (y | do(t)) =Xm

P (m | t)Xt0

P (y | m, t0 ) P (t0 )A set of variables M satisfies the frontdoor criterion relative to T and Y
if the following are true:
1. M completely mediates the effect of T on Y (i.e. all causal paths fromT to Y go through M).
2. There is no unblocked backdoor path from T to M.
3. All backdoor paths from M to Y are blocked by T.
Brady Neal

Frontdoor adjustment

21 / 40

See proof of frontdoor adjustment using
the truncated factorization in Section 6.1of the course bookQuestion:
What is the intuition for why the frontdoor
criterion gives us identifiability?

The magic of randomized experiments
Frontdoor adjustment
Pearl’s do-calculus
Determining identifiability from the graph

Brady Neal

Pearl’s do-calculus

24 / 40

Can we identify the causal effectif neither the backdoor criterion
nor the frontdoor criterion issatisfied?

Yes, and do-calculus tells us how.

Pearl’s do-calculus
Will allow us to identify any causal quantity that is identifiable

Brady Neal

Pearl’s do-calculus

27 / 40Pearl’s do-calculus
Will allow us to identify any causal quantity that is identifiable
P (Y | do(T = t), X = x)<latexit sha1_base64="PbGifmL7z2FVrihKI14Mtwy4tG4=">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</latexit>

Brady Neal

Pearl’s do-calculus

27 / 40Pearl’s do-calculus
Will allow us to identify any causal quantity that is identifiable
P (Y | do(T = t), X = x)<latexit sha1_base64="PbGifmL7z2FVrihKI14Mtwy4tG4=">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</latexit>

where Y, T, and X are arbitrary sets

Brady Neal

Pearl’s do-calculus

27 / 40Pearl’s do-calculus
Will allow us to identify any causal quantity that is identifiable
P (Y | do(T = t), X = x)<latexit sha1_base64="PbGifmL7z2FVrihKI14Mtwy4tG4=">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</latexit>

where Y, T, and X are arbitrary sets
Multiple treatments and/or multiple outcomes

Brady Neal

Pearl’s do-calculus

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Notation for Pearl’s do-calculus
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Notation for Pearl’s do-calculus
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Notation for Pearl’s do-calculus
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Rule 1 of do-calculus

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Pearl’s do-calculus

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P (y | do(t), z, w) = P (y | do(t), w)

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Pearl’s do-calculus

if Y ??GT Z | T, W

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Rule 1 of do-calculus<latexit sha1_base64="5aSrsPTDDmBuevDY57v75xBQ81M=">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</latexit>

P (y | do(t), z, w) = P (y | do(t), w)

if Y ??GT Z | T, W

Question: What concept does this remind you of ?

Brady Neal

Pearl’s do-calculus

29 / 40

Rule 1 of do-calculus<latexit sha1_base64="5aSrsPTDDmBuevDY57v75xBQ81M=">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</latexit>

P (y | do(t), z, w) = P (y | do(t), w)

if Y ??GT Z | T, W

Question: What concept does this remind you of ?

Brady Neal

Pearl’s do-calculus

29 / 40

Rule 1 of do-calculus<latexit sha1_base64="5aSrsPTDDmBuevDY57v75xBQ81M=">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</latexit>

P (y | do(t), z, w) = P (y | do(t), w)

if Y ??GT Z | T, W

Question: What concept does this remind you of ?
Rule 1 with do(t) removed:
P (y | z, w) = P (y | w)<latexit sha1_base64="djynxIpYlUCnIslXk0sHxpNh1Q4=">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</latexit>

Brady Neal

if Y ??G Z | W

Pearl’s do-calculus

29 / 40

Rule 1 of do-calculus<latexit sha1_base64="5aSrsPTDDmBuevDY57v75xBQ81M=">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</latexit>

P (y | do(t), z, w) = P (y | do(t), w)

if Y ??GT Z | T, W

Question: What concept does this remind you of ?
Rule 1 with do(t) removed:
P (y | z, w) = P (y | w)<latexit sha1_base64="djynxIpYlUCnIslXk0sHxpNh1Q4=">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</latexit>

if Y ??G Z | W

Generalization of d-separation to interventional distributions
Brady Neal

Pearl’s do-calculus

29 / 40

Rule 2 of do-calculus

Brady Neal

Pearl’s do-calculus

30 / 40

Rule 2 of do-calculus<latexit sha1_base64="HcYOjFqs6PGL84DcuD/U7PeV+Bc=">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</latexit>

P (y | do(t), do(z), w) = P (y | do(t), z, w)

Brady Neal

Pearl’s do-calculus

if Y ??GT ,Z Z | T, W

30 / 40

Rule 2 of do-calculus<latexit sha1_base64="HcYOjFqs6PGL84DcuD/U7PeV+Bc=">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</latexit>

P (y | do(t), do(z), w) = P (y | do(t), z, w)

if Y ??GT ,Z Z | T, W

Question: What concept does this remind you of ?

Brady Neal

Pearl’s do-calculus

30 / 40

Rule 2 of do-calculus<latexit sha1_base64="HcYOjFqs6PGL84DcuD/U7PeV+Bc=">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</latexit>

P (y | do(t), do(z), w) = P (y | do(t), z, w)

if Y ??GT ,Z Z | T, W

Question: What concept does this remind you of ?

Brady Neal

Pearl’s do-calculus

30 / 40

Rule 2 of do-calculus<latexit sha1_base64="HcYOjFqs6PGL84DcuD/U7PeV+Bc=">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</latexit>

P (y | do(t), do(z), w) = P (y | do(t), z, w)

if Y ??GT ,Z Z | T, W

Question: What concept does this remind you of ?
Rule 2 with do(t) removed:
P (y | do(z), w) = P (y | z, w)<latexit sha1_base64="2jPfRjazUWY+VS9eOWylbCzqCUM=">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</latexit>

Brady Neal

if Y ??GZ Z | W

Pearl’s do-calculus

30 / 40

Rule 2 of do-calculus<latexit sha1_base64="HcYOjFqs6PGL84DcuD/U7PeV+Bc=">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</latexit>

P (y | do(t), do(z), w) = P (y | do(t), z, w)

if Y ??GT ,Z Z | T, W

Question: What concept does this remind you of ?
Rule 2 with do(t) removed:
P (y | do(z), w) = P (y | z, w)<latexit sha1_base64="2jPfRjazUWY+VS9eOWylbCzqCUM=">AAATWnicrVjbbttGEF2lN8fpxYn71hc2ToEUkFXLDeAAhYEAdtKiaIAEdeI0oWFQFGUR5i0kZcch9Gl97jcU/ZmeObsUSVkXO4hkkcvZmTOXnRnuupcEfpZvbf3buvHJp599/sXKzdVbX3719Tdrt++8zOJR6nov3DiI01c9J/MCP/Je5H4eeK+S1HPCXuAd9k73ZP7wzEszP44O8ovEOwqdk8gf+K6Tg3S89o+d3LfjUe7GoXfmBJYd+n3L3o9BzIdeeubgL/ixbdluHA3iUdQHllCsXWumZE1qhpD9duSAKffe5YXlD6yxZf9ilSCW7YMz8XCJ8uPi1+PCpqi4Vrwej8cVulFWgz9e29jqbPFjXR50zWBDmc+z+PbK38pWfRUrV41UqDwVqRzjQDkqw/eN6qotlYB2pArQUox8zntqrFYhOwKXBw4H1FNcT/D0xlAjPAtmRmkXWgL8Ukha6gfD08d4QKq+i36rxjtPR0FssfEC957BDEHN1RDUZXIl51XlxKccFj6kLz7sTEgRL92GRwPcAzznsF+uF+D0MOpDKsXIBS0AVVNER4q7jqt4PmScHfJ5GIlN873pwfb5KyHzMb6nwHIwzmip2GqpJybqETV7tFV4Aq7YfMR38FDbtwhrMPHB56pqL4T3AJRT9R6jJvIinfXcmc+VG+TxDF3Cn5MjAncGSsy89hE/HxzNTHSBKTocruQJfUmYHx2D/Dvny1xNsJ6btCfjqlnMJp96ElM3ld2lnQHuPWKnkC8wN6QeiUUb9njETpmbZRTb5E7xdM4nlza6U/QO607Wsw2fJLvawI0x40+wdCR0PEuLtHUFaJb5bqo/qN/DurcZFcnrNmRjWqizOKOuyKz9LrqGxCXEVagS2ZLSpo6QdgyZLaegVfo0kuj7CV5krJ/AoBa4B4xaQpQ2tUt8zjkO6ZOs6Ii6zzHuU49IW+hkHbVj7Bg3tEpsfPaey1ot5p/Uru5701pFLmKlWiYrd6lpSz0wHssKiN31qLmm1+mOm9G78ZRVPdpfdQ3J0aZ9IlWtT3Ot+szGIaWauBLF2X7O8mWbddLmWkrsT8CxS6mBoWeszcRoWq3V356pkpjr4fIempzQ9S7azqdmCtBDE6kAfLrO5Y0jM2+B6TCnbEavPr8I8bGRL3t9D9+C1MVyL1lZTdkUo2Iys1je58hjfPRIVlzjJYx1Ql++n/ys2sxVsT8OXnQJUTI2N0hXRe8zN2Z7fqA20As2kFXNeMpKCyXl3iPB+N6E8x5whfeUWiJQpMKLyfx46RrsgzI2ERtwrqzDygafna9PTttknubQFZVPcUhVL9YqeTkrbzR9sWx/ku9NWU1fLCuUM+amb/pBRqxXS+RyroNIhLUsPVgiFfOdoOna5r+uYF/Ed+3I5EVqLDxcIhmyx2ivYlOVT5fal7MLpiYmIvP6AyJ4xv435g7ounGsZPNrRbOSu/igmFby59eMbCUZXjO+leT7JZIe3w3+hCYyz5fWlHA9W4r8drIv0VXU5rWH6Mi7uG86jmXeI7La+q23y9pum3dY+RYrd59Zrb/51F7tTAvuIs5NJyz3E5vUpH37DbX0FO9oGe+pn7E/6OL6pNE1r4oq+/SRqbc67jaQd9Dv9tkXr4+bmN30ZmPP3dTxGBrK39icTiruwOwh5yFdR+9sZJ3r/Ut72WlNercyBJfewV8sQayqZzbe4vWdjazP0TF3kNU+/mMg188oV8Orcmb+iXLZOXf+6SvBfYBrwI6QGd59Zl3AzuCpF+aEqN++e5fOupK/26gKyd3i7p0pfU386dPTCU8/I3BUPHoXnFNa0+qI885hCWVys9vIeGapTmadxklL403bltXyKjR4u6ZbeaYrrR6vbXSn/zdzeXC43ek+6HS7zx9sPHpo/nGzor5Td9V9BGpHPUJpPkNg3dZm68+W3Tq68996a/3m+i3NeqNlZNZV47P+7f8kDdwT</latexit>

if Y ??GZ Z | W

Generalization of backdoor adjustment/criterion
Brady Neal

Pearl’s do-calculus

30 / 40

Rule 3 of do-calculus

Brady Neal

Pearl’s do-calculus

31 / 40

Rule 3 of do-calculus<latexit sha1_base64="hzlHMvtn50jxkDa9AgCyCEidUMc=">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</latexit>

P (y | do(t), do(z), w) = P (y | do(t), w)

if Y ??GT ,Z(W ) Z | T, W

where Z(W ) denotes the set ofnodes of Z that aren't ancestorsof any node of W in GT<latexit 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Brady Neal

Pearl’s do-calculus

31 / 40

Rule 3 of do-calculus<latexit sha1_base64="hzlHMvtn50jxkDa9AgCyCEidUMc=">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</latexit>

P (y | do(t), do(z), w) = P (y | do(t), w)

if Y ??GT ,Z(W ) Z | T, W

where Z(W ) denotes the set ofnodes of Z that aren't ancestorsof any node of W in GT<latexit 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Brady Neal

Pearl’s do-calculus

31 / 40

Rule 3 of do-calculus<latexit sha1_base64="hzlHMvtn50jxkDa9AgCyCEidUMc=">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</latexit>

P (y | do(t), do(z), w) = P (y | do(t), w)

if Y ??GT ,Z(W ) Z | T, W

where Z(W ) denotes the set ofnodes of Z that aren't ancestorsof any node of W in GT<latexit 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Rule 3 with do(t) removed:<latexit sha1_base64="Cz0ARHQNlyRNX8egI5/3qu7lysM=">AAATTnicrVjbbttGEF2nN9e9Oc5jX5g4BRxAViXXgAMUBgLYSYqiAWzAjtNEgUFRlEWYN5OUHYfQlxX9hj73Q/pYtGfOLkVRVzuIZJHL2Zkzl50Z7rod+16aNRp/L9355NPPPv9i+cuVr77+5tvvVu+uvUyjfuK4x07kR8mrtp26vhe6x5mX+e6rOHHtoO27J+3zPZk/uXST1IvCo+w6dt8G9lnodT3HzkA6Xf2jFW+0on7mRIF7aftWK/A6Vms/AjHrucmljT//Uc1qOVHYjfphB1hCsXatqZLjbK2Lvg1y5r7LcsvrWgOr9bNViFktD5yxi0uYnebPT/NWBGPFl/z1xsmjwQDshR2TCk5X1xv1Bj/W5KBpBuvKfA6iu8t/qpbqqEg5qq8C5apQZRj7ylYpvm9UUzVUDNpblYOWYORx3lUDtQLZPrhccNignuN6hqc3hhriWTBTSjvQ4uOXQNJSPxieDsZdUvVd9FsjvLN05MQWG69xbxvMANRM9UBdJFdw3lROfMpg4WP64sHOmBTx0ql41MXdx3MG++V6DU4Xow6kEowc0HxQNUV0JLjruIrnPcbZJp+Lkdg025s2bJ+9EjIf4XsOLBvjlJaKrZZ6ZqIeUrNLW4XH54rNRnwHD7V987C6Qx88rqr2QniPQDlX7zGqIs/TOZo7s7kygzyYokv4M3KE4E5BiZjXHuLngaOaiQ4wRYfNlTyjLzHzo26Qf+V8kasx1nOT9qRcNYvZ5FFPbOqmtLuw08e9TewE8jnmetQjsajBHpfYCXOziGKN3Amervjk0EZnjF5n3cl61uCTZFcNuBFmvCGWjoSOZ2GRti4HzTLfTfUb9btY9xqjInldg2xEC3UWp9QVmrXfRdeQuAS4ClUiW1Bq1BHQjh6z5Ry0Up9GEn0/wouU9eMb1Bx3n1GLiVKjdonPFccBfZIV7VP3FcYd6hFpC52srnaMHYOKVomNx94zqdVi/knt6r43rlXkQlaqZbJyl5oaatt4LCsgdo9GzTG9TnfclN4Nxqxq0/6ya0iOVu0TqXJ9qmvVYTb2KFXFlShO93OaL1uskxrXUmJ/Bo5dSnUNPWVtxkbTykj97ZkqibgeDu+ByQld76LtamwmBz0wkfLBp+tc3jgycwFMmznVYvRG5+chPjXyRa9v45uTOl/uJSurKptglA9n5st7HLmMjx7Jimu8mLGO6cv94c8ambkp9sfBCycQJWMzg3RT9A5zY7rnR2odvWAdWVWNp6y0UBLuPWKMHw45HwJXeM+pJQRFKjwfzg8WrsE+KAMTsS7nijosbfDY+TrkbJnM0xy6orIxDqnq+VolL6fljabPl+0M870qq+nzZYVyydz0TD9IifVqgVzGdRCJYCRLjxZIRXwnaLq2+fcb2BfyXds3eZEYC08WSAbsMdqryFTli4X2ZeyCiYmJyLz+gAhesv8NuAO6bRxL2exW0Szlrj8opqX81S0jW0oGt4xvKfl+gaTLd4M3pInM4cKaEq6DhcgXw32JrqIar21ER97FHdNxLPMekdXWb71d1nbNvMOKt1ix+0xH+ptH7eXONOcu4sp0wmI/sUlN2rdfUEsv8I6W8Z76CfuDJq7PKl3zpqiyT++behvF3QLyDvrdPvvi7XFjs5verOy5qzqeQkPxG5jTScntmz3kLKTb6J2OrHO9M7GXHdekdys9cOkd/PUCxLJ6puPNX9/pyPocHXEHWe7jPwby6BnlZnhlzsw+US46584+fcW4d3H12RFSw7vPrPPZGVx1bE6I+u27N3HWlfzdQlVI7uYP1sb0VfHHT09nPP30wVHy6F1wRmlNG0WcdQ6LKZOZ3UbKM0t5MqtXTloab9y2dCSvAoO3a7qVa7rSyunqenP8fzOTg5OtenO73mwebq8/eWz+cbOsvlcP1AYCtaOeoDQPEFhn6f7S86WDpcO1v9b+Wft37T/NemfJyNxTlc+95f8BdOjZiA==</latexit>

P (y | do(z), w) = P (y | w)

Brady Neal

if Y ??GZ(W ) Z | W

Pearl’s do-calculus

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Rule 3 of do-calculus<latexit sha1_base64="hzlHMvtn50jxkDa9AgCyCEidUMc=">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</latexit>

P (y | do(t), do(z), w) = P (y | do(t), w)

if Y ??GT ,Z(W ) Z | T, W

where Z(W ) denotes the set ofnodes of Z that aren't ancestorsof any node of W in GT<latexit 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Rule 3 with do(t) removed:<latexit sha1_base64="Cz0ARHQNlyRNX8egI5/3qu7lysM=">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</latexit>

P (y | do(z), w) = P (y | w)

if Y ??GZ(W ) Z | W<latexit sha1_base64="qyXst99xVrsm2N99XQ2JOq0r7bg=">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</latexit>

P (y | do(z), w) = P (y | w)
Brady Neal

if Y ??GZ Z | W
Pearl’s do-calculus

31 / 40

Rule 3 of do-calculus<latexit sha1_base64="hzlHMvtn50jxkDa9AgCyCEidUMc=">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</latexit>

P (y | do(t), do(z), w) = P (y | do(t), w)

if Y ??GT ,Z(W ) Z | T, W

where Z(W ) denotes the set ofnodes of Z that aren't ancestorsof any node of W in GT<latexit 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A

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ZW

Rule 3 with do(t) removed:<latexit sha1_base64="Cz0ARHQNlyRNX8egI5/3qu7lysM=">AAATTnicrVjbbttGEF2nN9e9Oc5jX5g4BRxAViXXgAMUBgLYSYqiAWzAjtNEgUFRlEWYN5OUHYfQlxX9hj73Q/pYtGfOLkVRVzuIZJHL2Zkzl50Z7rod+16aNRp/L9355NPPPv9i+cuVr77+5tvvVu+uvUyjfuK4x07kR8mrtp26vhe6x5mX+e6rOHHtoO27J+3zPZk/uXST1IvCo+w6dt8G9lnodT3HzkA6Xf2jFW+0on7mRIF7aftWK/A6Vms/AjHrucmljT//Uc1qOVHYjfphB1hCsXatqZLjbK2Lvg1y5r7LcsvrWgOr9bNViFktD5yxi0uYnebPT/NWBGPFl/z1xsmjwQDshR2TCk5X1xv1Bj/W5KBpBuvKfA6iu8t/qpbqqEg5qq8C5apQZRj7ylYpvm9UUzVUDNpblYOWYORx3lUDtQLZPrhccNignuN6hqc3hhriWTBTSjvQ4uOXQNJSPxieDsZdUvVd9FsjvLN05MQWG69xbxvMANRM9UBdJFdw3lROfMpg4WP64sHOmBTx0ql41MXdx3MG++V6DU4Xow6kEowc0HxQNUV0JLjruIrnPcbZJp+Lkdg025s2bJ+9EjIf4XsOLBvjlJaKrZZ6ZqIeUrNLW4XH54rNRnwHD7V987C6Qx88rqr2QniPQDlX7zGqIs/TOZo7s7kygzyYokv4M3KE4E5BiZjXHuLngaOaiQ4wRYfNlTyjLzHzo26Qf+V8kasx1nOT9qRcNYvZ5FFPbOqmtLuw08e9TewE8jnmetQjsajBHpfYCXOziGKN3Amervjk0EZnjF5n3cl61uCTZFcNuBFmvCGWjoSOZ2GRti4HzTLfTfUb9btY9xqjInldg2xEC3UWp9QVmrXfRdeQuAS4ClUiW1Bq1BHQjh6z5Ry0Up9GEn0/wouU9eMb1Bx3n1GLiVKjdonPFccBfZIV7VP3FcYd6hFpC52srnaMHYOKVomNx94zqdVi/knt6r43rlXkQlaqZbJyl5oaatt4LCsgdo9GzTG9TnfclN4Nxqxq0/6ya0iOVu0TqXJ9qmvVYTb2KFXFlShO93OaL1uskxrXUmJ/Bo5dSnUNPWVtxkbTykj97ZkqibgeDu+ByQld76LtamwmBz0wkfLBp+tc3jgycwFMmznVYvRG5+chPjXyRa9v45uTOl/uJSurKptglA9n5st7HLmMjx7Jimu8mLGO6cv94c8ambkp9sfBCycQJWMzg3RT9A5zY7rnR2odvWAdWVWNp6y0UBLuPWKMHw45HwJXeM+pJQRFKjwfzg8WrsE+KAMTsS7nijosbfDY+TrkbJnM0xy6orIxDqnq+VolL6fljabPl+0M870qq+nzZYVyydz0TD9IifVqgVzGdRCJYCRLjxZIRXwnaLq2+fcb2BfyXds3eZEYC08WSAbsMdqryFTli4X2ZeyCiYmJyLz+gAhesv8NuAO6bRxL2exW0Szlrj8opqX81S0jW0oGt4xvKfl+gaTLd4M3pInM4cKaEq6DhcgXw32JrqIar21ER97FHdNxLPMekdXWb71d1nbNvMOKt1ix+0xH+ptH7eXONOcu4sp0wmI/sUlN2rdfUEsv8I6W8Z76CfuDJq7PKl3zpqiyT++behvF3QLyDvrdPvvi7XFjs5verOy5qzqeQkPxG5jTScntmz3kLKTb6J2OrHO9M7GXHdekdys9cOkd/PUCxLJ6puPNX9/pyPocHXEHWe7jPwby6BnlZnhlzsw+US46584+fcW4d3H12RFSw7vPrPPZGVx1bE6I+u27N3HWlfzdQlVI7uYP1sb0VfHHT09nPP30wVHy6F1wRmlNG0WcdQ6LKZOZ3UbKM0t5MqtXTloab9y2dCSvAoO3a7qVa7rSyunqenP8fzOTg5OtenO73mwebq8/eWz+cbOsvlcP1AYCtaOeoDQPEFhn6f7S86WDpcO1v9b+Wft37T/NemfJyNxTlc+95f8BdOjZiA==</latexit>

P (y | do(z), w) = P (y | w)

if Y ??GZ(W ) Z | W

W<latexit sha1_base64="qyXst99xVrsm2N99XQ2JOq0r7bg=">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</latexit>

P (y | do(z), w) = P (y | w)
Brady Neal

if Y ??GZ Z | W
Pearl’s do-calculusY

31 / 40

Rule 3 of do-calculus<latexit sha1_base64="hzlHMvtn50jxkDa9AgCyCEidUMc=">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</latexit>

P (y | do(t), do(z), w) = P (y | do(t), w)

if Y ??GT ,Z(W ) Z | T, W

where Z(W ) denotes the set ofnodes of Z that aren't ancestorsof any node of W in GT<latexit 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A

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ZW

Rule 3 with do(t) removed:<latexit sha1_base64="Cz0ARHQNlyRNX8egI5/3qu7lysM=">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</latexit>

P (y | do(z), w) = P (y | w)

if Y ??GZ(W ) Z | W

W<latexit sha1_base64="qyXst99xVrsm2N99XQ2JOq0r7bg=">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</latexit>

P (y | do(z), w) = P (y | w)
Brady Neal

if Y ??GZ Z | W
Pearl’s do-calculusY

31 / 40

Rule 3 of do-calculus<latexit sha1_base64="hzlHMvtn50jxkDa9AgCyCEidUMc=">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</latexit>

P (y | do(t), do(z), w) = P (y | do(t), w)

if Y ??GT ,Z(W ) Z | T, W

where Z(W ) denotes the set ofnodes of Z that aren't ancestorsof any node of W in GT<latexit 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A

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ZW

Rule 3 with do(t) removed:<latexit sha1_base64="Cz0ARHQNlyRNX8egI5/3qu7lysM=">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</latexit>

P (y | do(z), w) = P (y | w)

if Y ??GZ(W ) Z | W

W<latexit sha1_base64="qyXst99xVrsm2N99XQ2JOq0r7bg=">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</latexit>

P (y | do(z), w) = P (y | w)
Brady Neal

if Y ??GZ Z | W
Pearl’s do-calculusY

31 / 40

The rules of do-calculus
Rule 1: P (y | do(t), z, w) = P (y | do(t), w)<latexit sha1_base64="5aSrsPTDDmBuevDY57v75xBQ81M=">AAATenicrVjbbttGEF2lN9e9xLEf+8JEKeCgsiq5BhygMBDATloUNeAAdpw2DAyKWlmEeDO5suKw/LR+SL+gH9GXnpldiqKsm4NIFrmcnTlz2Znhrjux76Wq1fqndu+TTz/7/Iu1L9e/+vqbb+9vPNh8lUbDxJVnbuRHyeuOk0rfC+WZ8pQvX8eJdIKOL887g0OaP7+WSepF4am6ieXbwLkMvZ7nOgqki41/7XjbjobKjQJ57fiWHXhdyz6Ktm0FGBXIUIH8pGHZkerL5NrBn48nNwp70TDsAprmrQNrZaBpUftq6IBVyXcqs7yelVv2z1YBZdkeOGOJS6gusl8uMjuCP+RudprneWmX0TjWVtV0sVFvNVv8sW4P2mZQF+ZzEj1Y+1vYoisi4YqhCIQUoVAY+8IRKb5vRFu0RAzaW5GBlmDk8bwUuViH7BBcEhwOqANcL/H0xlBDPBNmytIutPj4JZC0xPeGp4txj6n6TvqtCd55OjLGJhtvcO8YzABUJfqgLpMrOFeVI58ULHzKvniwM2YKeelWPOrh7uNZwX663oBTYtSFVIKRC5oPqqaQjgR3HVfyvM9xdphPYkQ2zfemA9vnrwTNR/gOgOVgnLKlZKslXpioh6xZsq3E4/OKzUd8Bw+1fYuwemMfPF5V7QXxnoIyEO8xqiIv0jmZO/O5lEHOZ+gifsUcIbhTUCLOaw/x88BRzUQXmKTD4ZW8ZF9izo+mQf6N54tcjbGeO2xPyqtmcTZ5rCc2dVPaXdjp495h7ATyGeb6rIdi0YA9krETzs0iig3mTvA04ieXbXSn6E2uO1rPBnyi7GoAN8KMN8bSkdDxLCzS1mWgWea7I35n/RLr3uCoUF43IBuxhTqLU9YVmrU/QNeguAS4EpUiW1AarCNgO/qcLQPQSn0aifT9CC9Srh/foGa4+xy1mFEarJ3iM+JxwD7Rig5Z9wjjLushaQudrCn2jR15RSvFxuPec1urxflHtav73rRWkgu5Ui2TlQesqSX2jMe0AmT3ZNRc0+t0x03Zu3zKqg7bX3YNytGqfSRVrk91rbqcjX2WquJSFGf7OcuXXa6TBq8lxf4SHAcs1TP0lGszNprWJ+rv0FRJxOvh8j0wOaHrnbSNpmYy0AMTKR98us7pjUMzV8B0OKdsjt7k/CLE50a+6PUdfDOmLpZ7xZVVlU0wysYzi+U9HkmOjx7Rimu8mGMdsy8Pxz9rYmZV7I+DF95CpIxVBmlV9C7nxmzPT0UdvaCOrKrGk1aaKAnvPWKMH485HwOXeAesJQSFKjwbz+dL1+AIlNxErMdzRR2WNnjc+brMaZvM0xy6otQUB1X1Yq2Ul7PyRtMXy3bH+V6V1fTFskS55tz0TD9IGev1EjnF60ASwUSWni6RividoOna5j9WsC/kd+3Q5EViLDxfIhlwj9FeRaYqj5fap7gLJiYmJPPnB0Twmvtfzjugu8axlFV3imYpd/NBMS3lR3eMbCkZ3DG+peT7JZKS3w3emEYyL5fWFHGdLEW+Gu9LdBU1+NpBdOhd3DUdxzLvEVpt/dY74NpumHdY8RYrdp/pRH/zWHu5M814FzEynbDYT+ywJu3br6ilY7yjaXwofsL+oI3ri0rXXBWV9ulDU2+TuLtA3ke/O+K+eHfc2Oymdyp77qqO59BQ/HJzOim5fbOHnId0F72zkXWud2/tZac16d1KH1x6B3+zBLGsntl4i9d3NrI+R0e8gyz38R8DefKMshpemTPzT5TLzrnzT18x7j1cfe4IqeE94qzzuTNIcWZOiPrte3jrrEv5u4uqoNzNHm1O6aviT5+eLvn0MwRHyaN3wYqlNW0Scd45LGYZZXYbKZ9ZypNZs3LS0njTtqUTeRUYvAPTraTpSusXG/X29P9mbg/Od5vtvWa7/XKv/uyp+cfNmvhOPBLbCNS+eIbSPEFg3dpxLa39Vcs3/9t6uPVk6wfNeq9mZLZE5bO19z8Kfujv</latexit>

if Y ??GT Z | T, W

Rule 2: P (y | do(t), do(z), w) = P (y | do(t), z, w)

if Y ??GT ,Z Z | T, W

Rule 3: P (y | do(t), do(z), w) = P (y | do(t), w)

if Y ??GT ,Z(W ) Z | T, W

<latexit sha1_base64="HcYOjFqs6PGL84DcuD/U7PeV+Bc=">AAATnXicrVjbbttGEF2lt8S9xIkfCxRsnAIpIKuWGyABCgMB7KRFkRQJ4MRposCgKMoizFtIyo5D8Df62v5IP6R/0zNnlyIp6+YgEkQuZ2fOXHZmuKt+7Htptr39X+vKJ59+9vkXV6+tffnV199cX79x80UajRPHfe5EfpS87Nup63uh+zzzMt99GSeuHfR997B/sifzh6duknpReJCdx+6bwD4OvaHn2BlIRzda13rxnV40zpwocE9t3+oF3sDq7Ud3ehlwssANM5B/bGtalI3c5NRODMmJwmE0DgdQIBRr11oZrYY0A6j3dmxDMHPfZbnlDa3C6v1ilcBWzwNn7OISZkf5r0d5L4KPEoL8oAAagfj4qiiKSpcxZ2JKU/HR+uZ2Z5sf6+KgawabynyeRjeu/qt6aqAi5aixCpSrQpVh7Ctbpfi+Vl21rWLQ3qgctAQjj/OuKtQaZMfgcsFhg3qC6zGeXhtqiGfBTCntQIuPXwJJS/1geAYYD0nVd9Fv1Xjn6ciJLTae4943mAGomRqBukyu5FxVTnzKYOF9+uLBzpgU8dJpeDTE3cdzBvvleg5OF6MBpBKMHNB8UDVFdCS467iK5yPG2Safi5HYNN+bPmyfvxIyH+F7Aiwb45SWiq2WemSiHlKzS1uFx+eKzUd8Bw+1fYuwhhMfPK6q9kJ4D0A5Ue8xaiIv0lnPnflcmUEuZugS/owcIbhTUCLmtYf4eeBoZqIDTNFhcyWP6UvM/OgY5N85X+ZqjPXcoj0pV81iNnnUE5u6qewu7fRx7xM7gXyOuRH1SCzasMcldsLcLKPYJneCpzM+ObTRmaJ3WHeynm34JNnVBm6EGW+CpSOh41lapK3LQbPMd0s9pn4X695mVCSv25CNaKHO4pS6QrP2u+gaEpcAV6FKZEtKmzoC2jFitpyAVunTSKLvJ3iRsn58g5rj7jNqMVHa1C7xOeM4oE+yomPqPsN4QD0ibaGTddQ9Y0fR0Cqx8dh7Lmq1mH9Su7rvTWsVuZCVapms3KWmbXXXeCwrIHbXo+aYXqc7bkrviimr+rS/6hqSo037RKpan+ZaDZiNI0o1cSWKs/2c5csO66TNtZTYH4Njl1JDQ09Zm7HRtFarvz1TJRHXw+E9MDmh6120nU3N5KAHJlI++HSdyxtHZt4C02ZO9Ri9+vwixIdGvuz1fXxzUhfLvWBlNWUTjPLJzGJ5jyOX8dEjWXGNFzPWMX35fvKzajOrYn8cvPAComRsZpBWRR8wN2Z7fqA20Qs2kVXNeMpKCyXh3iPG+PaE8zZwhfeEWkJQpMLzyXyxdA32QSlMxIacK+uwssFj5xuQs2cyT3PoisqmOKSqF2uVvJyVN5q+WHYwyfemrKYvlhXKKXPTM/0gJdbLJXIZ10EkglqWHiyRivhO0HRt858r2BfyXTs2eZEYCw+XSAbsMdqryFTlk6X2ZeyCiYmJyLz6gAiesv8V3AFdNo6VbHapaFZy5x8U00r+7JKRrSSDS8a3kny/RNLlu8Gb0ETm2dKaEq6nS5HfTvYluoravPYRHXkXD0zHscx7RFZbv/V2Wdtt8w4r32Ll7jOt9TeP2qudac5dxJnphOV+YouatG+/oZae4B0t4z31M/YHXVwfNbrmqqiyTx+beqvj7gD5HvrdPvvi5XFjs5veauy5mzoeQkP5K8zppOL2zR5yHtJl9M5G1rk+uLCXndakdysjcOkd/PkSxKp6ZuMtXt/ZyPocHXEHWe3jPwZy/YyyGl6VM/NPlMvOufNPXzHuQ1x9doTU8O4z63x2Blc9NydE/fbdu3DWlfzdQVVI7ua3bk7pa+JPn56OefoZg6Pi0bvgjNKaVkecdw6LKZOZ3UbKM0t1Mus0Tloab9q2tJZXgcHbNd3KNV1p7Wh9szv938zFweFOp3u30+0+u7v54L754+aq+lbdUncQqHvqAUrzKQLrtOLWX62/W/9sfLexv/F44w/NeqVlZDZU47Nx+D/QFvTF</latexit><latexit sha1_base64="hzlHMvtn50jxkDa9AgCyCEidUMc=">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</latexit>

Brady Neal

Pearl’s do-calculus

32 / 40

The rules of do-calculus
Rule 1: P (y | do(t), z, w) = P (y | do(t), w)<latexit sha1_base64="5aSrsPTDDmBuevDY57v75xBQ81M=">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</latexit>

if Y ??GT Z | T, W

Rule 2: P (y | do(t), do(z), w) = P (y | do(t), z, w)

if Y ??GT ,Z Z | T, W

Rule 3: P (y | do(t), do(z), w) = P (y | do(t), w)

if Y ??GT ,Z(W ) Z | T, W

<latexit sha1_base64="HcYOjFqs6PGL84DcuD/U7PeV+Bc=">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</latexit><latexit sha1_base64="hzlHMvtn50jxkDa9AgCyCEidUMc=">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</latexit>

Proof of the frontdoor adjustment using do-calculus in Section 6.2.1 of thecourse book (compare with proof using truncated factorization in Section 6.1)

Brady Neal

Pearl’s do-calculus

32 / 40

Completeness of do-calculus
Maybe there are some identifiable causal estimands that can’t be identified
using the rules of do-calculus

Brady Neal

Pearl’s do-calculus

33 / 40

Completeness of do-calculus
Maybe there are some identifiable causal estimands that can’t be identified
using the rules of do-calculus
Fortunately, not, as do-calculus is complete (Shpitser & Pearl, 2006a; Huang& Valtorta, 2006; Shpitser & Pearl, 2006b)

Brady Neal

Pearl’s do-calculus

33 / 40

Completeness of do-calculus
Maybe there are some identifiable causal estimands that can’t be identified
using the rules of do-calculus
Fortunately, not, as do-calculus is complete (Shpitser & Pearl, 2006a; Huang& Valtorta, 2006; Shpitser & Pearl, 2006b)
Constructive proofs that admit polynomial time algorithms for
identification

Brady Neal

Pearl’s do-calculus

33 / 40

Question:
What concepts are the first and second
rules of do-calculus generalizations of ?

The magic of randomized experiments
Frontdoor adjustment
Pearl’s do-calculus
Determining identifiability from the graph

Brady Neal

Determining identifiability from the graph

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<latexit sha1_base64="tPu6QooQe9BHGcmEUBnYAaVnPGc=">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</latexit>

W1

Question:In this graph, is the
backdoor criterion satisfied?

W2

T

M1

M2

Y

<latexit sha1_base64="tPu6QooQe9BHGcmEUBnYAaVnPGc=">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</latexit>

W1

Question:In this graph, is the
backdoor criterion satisfied?
How about the frontdoor
criterion?

W2

T

M1

M2

Y

Unconfounded children criterion
This criterion is satisfied if it is possible
to block all backdoor paths from thetreatment variable T to all of its childrenthat are ancestors of Y with a single
conditioning set (Tian & Pearl, 2002).

Brady Neal

Determining identifiability from the graph

37 / 40

Unconfounded children criterion
This criterion is satisfied if it is possible
to block all backdoor paths from thetreatment variable T to all of its childrenthat are ancestors of Y with a single
conditioning set (Tian & Pearl, 2002).

Sufficient condition for identifiability
when T is a single variable

Brady Neal

Determining identifiability from the graph

37 / 40

Unconfounded children criterion
non-causal association<latexit sha1_base64="mU924qzy7v37q/Pq1lkmDU8AmZU=">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</latexit>

This criterion is satisfied if it is possible
to block all backdoor paths from thetreatment variable T to all of its childrenthat are ancestors of Y with a single
conditioning set (Tian & Pearl, 2002).

W1

W2

M1

T

Sufficient condition for identifiability
when T is a single variableYM2
causal association

Brady Neal

Determining identifiability from the graph

37 / 40

Unconfounded children criterion
non-causal association<latexit sha1_base64="66ys+si+DqrSdIev/ZBVuNabKGU=">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</latexit>

This criterion is satisfied if it is possible
to block all backdoor paths from thetreatment variable T to all of its childrenthat are ancestors of Y with a single
conditioning set (Tian & Pearl, 2002).

W1

W2

focus
M1

T

Sufficient condition for identifiability
when T is a single variableYM2
causal association

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Unconfounded children criterion
This criterion is satisfied if it is possible
to block all backdoor paths from thetreatment variable T to all of its childrenthat are ancestors of Y with a single
conditioning set (Tian & Pearl, 2002).

<latexit sha1_base64="b+Ugtbuwp7PB0umDIM5NBagzCGU=">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</latexit>

W1

W2

focus
M1

T

Sufficient condition for identifiability
when T is a single variableYM2
causal association

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Necessary condition for identifiability
For each backdoor path from T to any child M of T that is an ancestor ofY, it is possible to block that path (Pearl, 2009, p. 92).

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Necessary condition for identifiability
For each backdoor path from T to any child M of T that is an ancestor ofY, it is possible to block that path (Pearl, 2009, p. 92).
<latexit sha1_base64="qoKSZFy6ZA8ulX3Xr8JW92Lh8Kg=">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</latexit>

W1

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T

Brady NealY

Determining identifiability from the graph

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Necessary and sufficient condition

Brady Neal

Determining identifiability from the graph

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Necessary and sufficient condition
Recall: identification with the rules do-calculus is necessary and sufficient(Shpitser & Pearl, 2006a; Huang & Valtorta, 2006; Shpitser & Pearl, 2006b)

Brady Neal

Determining identifiability from the graph

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Necessary and sufficient condition
Recall: identification with the rules do-calculus is necessary and sufficient(Shpitser & Pearl, 2006a; Huang & Valtorta, 2006; Shpitser & Pearl, 2006b)
For graphical criterion, see Shpitser & Pearl, 2006a, 2006b

Brady Neal

Determining identifiability from the graph

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Necessary and sufficient condition
Recall: identification with the rules do-calculus is necessary and sufficient(Shpitser & Pearl, 2006a; Huang & Valtorta, 2006; Shpitser & Pearl, 2006b)
For graphical criterion, see Shpitser & Pearl, 2006a, 2006b: hedge criterion

Brady Neal

Determining identifiability from the graph

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Necessary and sufficient condition
Recall: identification with the rules do-calculus is necessary and sufficient(Shpitser & Pearl, 2006a; Huang & Valtorta, 2006; Shpitser & Pearl, 2006b)
For graphical criterion, see Shpitser & Pearl, 2006a, 2006b: hedge criterion

Brady Neal

Determining identifiability from the graph

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Necessary and sufficient condition
Recall: identification with the rules do-calculus is necessary and sufficient(Shpitser & Pearl, 2006a; Huang & Valtorta, 2006; Shpitser & Pearl, 2006b)
For graphical criterion, see Shpitser & Pearl, 2006a, 2006b: hedge criterion

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Determining identifiability from the graph

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Questions:
1. Is the unconfounded children criterion satisfied here?

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W1

W3

W2

T

M

Y

Questions:
1. Is the unconfounded children criterion satisfied here?
2. How about here?

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Questions:
1. Is the unconfounded children criterion satisfied here?
2. How about here?
3. Can we get identifiability via any simpler criterion that
we've seen before?
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