Estimation
Brady Nealcausalcourse.com

Estimation portion of the flowchart<latexit sha1_base64="h8jyz1uHnGldXgp26oClVqsQHYc=">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</latexit>

Causal Estimand

Brady Neal

Identification

Statistical Estimand

Estimation

Estimate

2 / 35Preliminaries
Conditional average treatment effects (CATEs):<latexit sha1_base64="Ctzs3z4aY6VPZUhbj4ydOdQEuFI=">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</latexit>

Brady Neal

⌧ (x) , E[Y (1)Y (0) | X = x]

3 / 35Preliminaries
Conditional average treatment effects (CATEs):<latexit sha1_base64="Ctzs3z4aY6VPZUhbj4ydOdQEuFI=">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</latexit>⌧ (x) , E[Y (1)Y (0) | X = x]

Always assuming unconfoundedness and positivity

Brady Neal3 / 35Preliminaries
Conditional average treatment effects (CATEs):<latexit sha1_base64="Ctzs3z4aY6VPZUhbj4ydOdQEuFI=">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</latexit>⌧ (x) , E[Y (1)Y (0) | X = x]

Always assuming unconfoundedness and positivity⌧ , E[Y (1)<latexit sha1_base64="nkm/C9Jn2NA5cqfAGWnaB02y2ao=">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</latexit>

Y (0)] = EW [E[Y | T = 1, W ]E[Y | T = 0, W ]]Given W is a sufficient adjustment set<latexit sha1_base64="y6nnSigj2UkfW7tvC9oZQnFjROI=">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</latexit>

Brady Neal3 / 35Preliminaries
Conditional average treatment effects (CATEs):<latexit sha1_base64="Ctzs3z4aY6VPZUhbj4ydOdQEuFI=">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</latexit>⌧ (x) , E[Y (1)Y (0) | X = x]

Always assuming unconfoundedness and positivity⌧ , E[Y (1)<latexit sha1_base64="nkm/C9Jn2NA5cqfAGWnaB02y2ao=">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</latexit>

Y (0)] = EW [E[Y | T = 1, W ]E[Y | T = 0, W ]]Given W is a sufficient adjustment set<latexit sha1_base64="y6nnSigj2UkfW7tvC9oZQnFjROI=">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</latexit>⌧ (x) , E[Y (1)<latexit sha1_base64="KWSzAeHNia9cnBsm5+I3e5sEX8M=">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</latexit>

Y (0) | X = x] = EW [E[Y | T = 1, X = x, W ]E[Y | T = 0, X = x, W ]]Given W [ X is a sufficient adjustment set<latexit sha1_base64="HJoRYiEmPGuhM+q3HZJwgKHYVJI=">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</latexit>

Brady Neal3 / 35

Conditional Outcome Modeling
Increasing Data Efficiency
Propensity Scores and IPW
Other Methods

Brady Neal

4 / 35

Conditional Outcome Modeling
Increasing Data Efficiency
Propensity Scores and IPW
Other Methods

Brady Neal

Conditional Outcome Modeling5 / 35

Conditional outcome modeling (COM)⌧ = EW [E[Y | T = 1, W ]<latexit sha1_base64="6c1Bz+GSJyXbD1v3ABdCutoCeLI=">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</latexit>

Brady NealE[Y | T = 0, W ]]Conditional Outcome Modeling6 / 35

Conditional outcome modeling (COM)⌧ = EW [E[Y | T = 1, W ]<latexit sha1_base64="6c1Bz+GSJyXbD1v3ABdCutoCeLI=">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</latexit>

model

Brady NealE[Y | T = 0, W ]]model

Conditional Outcome Modeling6 / 35

Conditional outcome modeling (COM)⌧ = EW [E[Y | T = 1, W ]<latexit sha1_base64="6c1Bz+GSJyXbD1v3ABdCutoCeLI=">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</latexit>

model

Brady NealE[Y | T = 0, W ]]model

Conditional Outcome Modeling6 / 35

Conditional outcome modeling (COM)⌧ = EW [E[Y | T = 1, W ]<latexit sha1_base64="6c1Bz+GSJyXbD1v3ABdCutoCeLI=">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</latexit>

E[Y | T = 0, W ]]

model

model

⌧ = EW [µ(1, W )µ(0, W )]<latexit sha1_base64="sDoQAtycoL6a2NeSyENAcLTzMto=">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</latexit>

Brady Neal

Conditional Outcome Modeling6 / 35

Conditional outcome modeling (COM)⌧ = EW [E[Y | T = 1, W ]<latexit sha1_base64="6c1Bz+GSJyXbD1v3ABdCutoCeLI=">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</latexit>

E[Y | T = 0, W ]]

model

model

⌧ = EW [µ(1, W )µ(0, W )]<latexit sha1_base64="sDoQAtycoL6a2NeSyENAcLTzMto=">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</latexit>

Model-assisted estimator:1X⌧ˆ =(µ̂(1, wi )n i<latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit>

Brady Nealµ̂(0, wi ))Conditional Outcome Modeling6 / 35

COM estimation of CATEs
1X
ATE COM Estimator: ⌧ˆ =(µ̂(1, wi )n i<latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit>

Brady Nealµ̂(0, wi ))Conditional Outcome Modeling7 / 35

COM estimation of CATEs
1X
ATE COM Estimator: ⌧ˆ =(µ̂(1, wi )n i<latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit>µ̂(0, wi ))CATE Estimand:⌧ (x) , E[Y (1)<latexit sha1_base64="KWSzAeHNia9cnBsm5+I3e5sEX8M=">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</latexit>

Brady Neal

Y (0) | X = x] = EW [E[Y | T = 1, X = x, W ]Conditional Outcome ModelingE[Y | T = 0, X = x, W ]]7 / 35

COM estimation of CATEs
1X
ATE COM Estimator: ⌧ˆ =(µ̂(1, wi )n i<latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit>µ̂(0, wi ))CATE Estimand:⌧ (x) , E[Y (1)<latexit sha1_base64="KWSzAeHNia9cnBsm5+I3e5sEX8M=">AAATh3icrVjbbttGEF2nN8e9OfFjX9g4BRxAViQ3jfNiIIXjoCgaIAGcOKllGBRF2YR5M7my6wh66Gv7Tf2Qfkjfe+bsSiRlXawgEkQuZ2fOzM6Nu2qnYZDrRuPfpVuffPrZ518s31758quvv/l29c7dN3nSyzz/tZeESfa27eZ+GMT+ax3o0H+bZr4btUP/oH22K/MHF36WB0m8r69S/yhyT+KgG3iuBul49b+WdnsbLS+5cLPA1X5+4YYPnJbGQ3wS+udOa+/w3UbzgbPpvNtoYCYKOk6J3dlxqsJHQtk7PjiEYCvpaS+JfCulYZiO/FiDpVmbicLZuJv04g6MP4L22XCNheCOjlfXG/UGP871QdMO1pX9vEzuLP+jWqqjEuWpnoqUr2KlMQ6Vq3J8D1VTNVQK2pHqg5ZhFHDeVwO1AtkeuHxwuKCe4XqCp0NLjfEsmDmlPWgJ8csg6agfLE8H4y6p5i76nRLvNB19YouNV7i3LWYEqlanoM6TG3LeVE7WpGHhE64lgJ0pKbJKr7KiLu4hnjXsl+sVOH2MOpDKMPJAC0E1FNGR4W78Kis/pZ9d8vkYiU3TV9OG7dMjIfMJvmfAcjHOaanY6qjn1usxNfu0VXhCRmw64h9YobFvFlZ3tIaAUTWrEN59UM7Ue4yqyLN0lnNnOpe2yIMJuoRfkyMGdw5KwrwO4L8AHNVM9IApOlxG8oRrSZkfdYv8K+eHuZoinpu0J2fUHGZTQD2prZvC7qGdIe5tYmeQ72PulHrEFzXY4xM7Y24OvVgjd4anSz55tNEbo9dZdxLPGtYk2VUDboKZYIRlPGH8ObTIWNcHzbHfTfUb9fuIe41ekbyuQTahhSaLc+qKbex30DXELxGuQhXPDik16ohoxymz5Qy0Qp9BEn0PsYqc9RNa1D7uIb2WEqVG7eKfS44jrkki2qPuS4w71CPSDjpZXW1bOwYVreKbgL3nulaH+Se1a/reuFaRi1mpjs3KHWpqqEd2xRIBsbvsNc/2OtNxc65uMGZVm/YXXUNytGqfSBXxqcaqw2w8pVQVV7w4eZ2T1rLFOqkxluL7E3DsUKpr6TlrM7WaVkr1t2urJGE8PN4jmxOm3kXb5dhMH/TIeioEn6lzeePIzDkwXeZUi94rz89C3LPyw17fxrdP6my5N6ysqmyGUX80M1s+4Minf8xIIm7wUvo65Vq+H/2c0sxNsT8OXnwNUTJWW6SboneYG5NXvq/W0QvWkVVVf0qkhZJx75FifH/EeR+4wntGLTEoUuH90fxgbgyegTKwHutybliHhQ0BO1+HnC2beYbDVJQe45Cqnq1V8nJS3hj6bNnOKN+rsoY+W1YoF8zNwPaDnFhv58hpxkEkolKW7s+RSvhOMHRj87sb2BfzXduzeZFZCw/mSEbsMWZVia3KF3Pt0+yCmfWJyPz+AR68YP8bcAe0qB8LWb2QNwu5qw/yaSF/uaBnC8loQf8Wku/nSPp8NwQjmsi8mltTwvVyLvL5aF9iqqjGaxvekXdxx3Ycx75HJNrmrbfD2q7Zd9jwLTbcfeal/hZQe7Ez7XMXcWk74XA/sUlNZm2/oJZe4B0t4131I/YHTVyfV7rmTVFln96z9VbG3QLyNvrdM/bFxXFTu5verOy5qzr2oGH4G9jTScEd2j3kNKRF9E5GNrneubaXHddkdiun4DI7+Ks5iEX1TMabHd/JyOYcnXAHWezjPwZy+YxyM7wiZ6afKOedc6efvlLcu7iG7Ai55X3GrAvZGXz12p4Qzdt399pZV/J3C1Uhudu/d3dMXxV//PR0wtNPDxwFj9kFa0obWhlx2jkspYy2u42cZ5biZFavnLQM3rhteSmvIou3Y7uVb7vSyvHqenP8v5nrgzdb9eZP9carR+tPH9v/bZbVd+qe2oCfttVTVOZL+NVbOlz6c+mvpb/Xbq89XHu89sSw3lqyMmuq8ln7+X+KTeip</latexit>

Y (0) | X = x] = EW [E[Y | T = 1, X = x, W ]E[Y | T = 0, X = x, W ]]µ(t, w, x) , E[Y | T = t, W = w, X = x]<latexit sha1_base64="NUn2yjqlAs8cjD5+3hs3RYaw7yo=">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</latexit>

Brady Neal

Conditional Outcome Modeling7 / 35

COM estimation of CATEs
1X
ATE COM Estimator: ⌧ˆ =(µ̂(1, wi )n i<latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit>µ̂(0, wi ))CATE Estimand:⌧ (x) , E[Y (1)<latexit sha1_base64="KWSzAeHNia9cnBsm5+I3e5sEX8M=">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</latexit>

Y (0) | X = x] = EW [E[Y | T = 1, X = x, W ]E[Y | T = 0, X = x, W ]]µ(t, w, x) , E[Y | T = t, W = w, X = x]<latexit sha1_base64="NUn2yjqlAs8cjD5+3hs3RYaw7yo=">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</latexit>

CATE COM
Estimator:
Brady Neal

1 X
⌧ˆ(x) =(µ̂(1, wi , x)nx i:x =x<latexit sha1_base64="9aLqR6TNFSSe+UWpg2Abyn9KioQ=">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</latexit>µ̂(0, wi , x))i

Conditional Outcome Modeling7 / 35

<latexit sha1_base64="9Zf9BmLnXQJnoqhOEu7stCDJKjM=">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</latexit>⌧ˆi = ⌧ˆ(xi ) = µ̂(1, wi , xi )µ̂(0, wi , xi )Question:
What could go wrong with this estimator?

COM estimation’s many faces

Brady Neal

Conditional Outcome Modeling9 / 35

COM estimation’s many faces• G-computation estimators

Brady Neal

Conditional Outcome Modeling9 / 35

COM estimation’s many faces• G-computation estimators

Brady Neal

Conditional Outcome Modeling9 / 35

COM estimation’s many faces• G-computation estimators• Parametric G-formula

Brady Neal

Conditional Outcome Modeling9 / 35

COM estimation’s many faces• G-computation estimators• Parametric G-formula

Brady Neal

Conditional Outcome Modeling9 / 35

COM estimation’s many faces• G-computation estimators• Parametric G-formula• Standardization

Brady Neal

Conditional Outcome Modeling9 / 35

COM estimation’s many faces• G-computation estimators• Parametric G-formula• Standardization• S-learner where “S” is for “Single”

Brady Neal

Conditional Outcome Modeling9 / 35

Problem with COM estimation in high dimensions1X⌧ˆ =(µ̂(1, wi )n i<latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit>

Brady Neal

Conditional Outcome Modelingµ̂(0, wi ))10 / 35

Problem with COM estimation in high dimensions1X⌧ˆ =(µ̂(1, wi )n i<latexit 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wi ))<latexit 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W<latexit sha1_base64="VPvP5OxvTxlPGVcROXV6+AtJi24=">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</latexit>

Brady Neal

T
W

Y

Conditional Outcome Modeling

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Problem with COM estimation in high dimensions1X⌧ˆ =(µ̂(1, wi )n i<latexit 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wi ))<latexit 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W<latexit sha1_base64="VPvP5OxvTxlPGVcROXV6+AtJi24=">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</latexit>

Brady Neal

T
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Conditional Outcome Modeling

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Problem with COM estimation in high dimensions1X⌧ˆ =(µ̂(1, wi )n i<latexit 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wi ))<latexit 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W<latexit sha1_base64="VPvP5OxvTxlPGVcROXV6+AtJi24=">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</latexit>

Brady Neal

T
W

Y

Conditional Outcome Modeling

Problem: estimate canbe biased toward zero
(Künzel et al., 2019)10 / 35

How can we ensure that themodel doesn’t ignore T?

Grouped COM (GCOM) estimation1X(µ̂(1, wi )COM: ⌧ˆ =n i<latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit>

Brady Nealµ̂(0, wi ))Conditional Outcome Modeling

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Grouped COM (GCOM) estimation1X(µ̂(1, wi )COM: ⌧ˆ =n i<latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit>1X(µ̂1 (wi )GCOM: ⌧ˆ =n i

µ̂(0, wi ))<latexit sha1_base64="ls0sKV6pNN44O/vMX7/T0OvDurI=">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</latexit>

Brady Nealµ̂0 (wi ))Conditional Outcome Modeling

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Grouped COM (GCOM) estimation<latexit sha1_base64="2blVoQ+k5PyAuiF7BR/rXP61udw=">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</latexit>1X(µ̂(1, wi )COM: ⌧ˆ =n i<latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit>1X(µ̂1 (wi )GCOM: ⌧ˆ =n i

µ̂(0, wi ))TWY<latexit sha1_base64="ls0sKV6pNN44O/vMX7/T0OvDurI=">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</latexit>

Brady Nealµ̂0 (wi ))Conditional Outcome Modeling

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Grouped COM (GCOM) estimation<latexit sha1_base64="2blVoQ+k5PyAuiF7BR/rXP61udw=">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</latexit>1X(µ̂(1, wi )COM: ⌧ˆ =n i<latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit>1X(µ̂1 (wi )GCOM: ⌧ˆ =n i

µ̂(0, wi ))TWY<latexit sha1_base64="ls0sKV6pNN44O/vMX7/T0OvDurI=">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</latexit>µ̂0 (wi ))T = 1 network

T = 0 network<latexit sha1_base64="8jMDGluqeHMSJvwUG4K/zbsc1Ow=">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</latexit><latexit sha1_base64="8jMDGluqeHMSJvwUG4K/zbsc1Ow=">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</latexit>

W

Brady NealYW

Conditional Outcome ModelingY

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Trained with treatment group data

Trained with control group data

T = 1 network

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Brady NealYW

Conditional Outcome ModelingY

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Trained with treatment group data

Trained with control group data

T = 1 network

T = 0 network<latexit sha1_base64="8jMDGluqeHMSJvwUG4K/zbsc1Ow=">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</latexit><latexit sha1_base64="8jMDGluqeHMSJvwUG4K/zbsc1Ow=">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</latexit>

W

Y

W

Y

Problem: networks have higher variance than they would
if they were trained with all the data (not efficient)
Brady Neal

Conditional Outcome Modeling

13 / 35

Question:
Write down the general form of a COM
estimator and a GCOM estimator.

Conditional Outcome Modeling
Increasing Data Efficiency
Propensity Scores and IPW
Other Methods

Brady Neal

Increasing Data Efficiency

15 / 35TARNetBrady Neal

Increasing Data Efficiency

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TW

Brady NealY

Increasing Data Efficiency

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TWY

Too biased!

Brady Neal

Increasing Data Efficiency

16 / 35TARNetGCOMCOM

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T
W

Y

W

T = 0 network

Too biased!

W

Brady NealY

Y

Increasing Data Efficiency

16 / 35TARNetGCOMCOM

T = 1 network<latexit sha1_base64="txXJgFDQnPuQf6d7IIXXvEg2lQk=">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</latexit><latexit sha1_base64="LSZwUq8CzOLCijicp5leilkvu/E=">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</latexit>

T
W

Y

W

Y

T = 0 network
Too much variance!

W

Brady NealY

Increasing Data Efficiency

16 / 35TARNetGCOM<latexit sha1_base64="m4is/x1qKs6D+47izt0tLlAKSRI=">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</latexit>

COM

TARNet(Shalit et al., 2017)

T = 1 network<latexit sha1_base64="txXJgFDQnPuQf6d7IIXXvEg2lQk=">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</latexit>

Y<latexit sha1_base64="LSZwUq8CzOLCijicp5leilkvu/E=">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</latexit>

1

Y

=

Y

W

W

T

T
W

T
=
0

T = 0 network
Too much variance!

YW

Brady NealY

Increasing Data Efficiency

16 / 35TARNetGCOM<latexit sha1_base64="m4is/x1qKs6D+47izt0tLlAKSRI=">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</latexit>

COM

TARNet(Shalit et al., 2017)

T = 1 network<latexit sha1_base64="txXJgFDQnPuQf6d7IIXXvEg2lQk=">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</latexit>

Y<latexit sha1_base64="LSZwUq8CzOLCijicp5leilkvu/E=">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</latexit>

1

Y

=

Y

W

W

T

T
W

T
=
0

T = 0 network
Just right!

W

Brady NealY

Y

Increasing Data Efficiency

16 / 35TARNetGCOM<latexit sha1_base64="m4is/x1qKs6D+47izt0tLlAKSRI=">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</latexit>

COM

TARNet(Shalit et al., 2017)

T = 1 network<latexit sha1_base64="txXJgFDQnPuQf6d7IIXXvEg2lQk=">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</latexit>

Y<latexit sha1_base64="LSZwUq8CzOLCijicp5leilkvu/E=">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</latexit>

1

Y

=

Y

W

W

T

T
W

T
=
0

T = 0 network
Y
W

Used in a COM
estimator:

Y

1X
⌧ˆ =
(µ̂(1, wi )n i<latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit>

Brady Neal

Increasing Data Efficiencyµ̂(0, wi ))16 / 35TARNet inefficiency<latexit sha1_base64="m4is/x1qKs6D+47izt0tLlAKSRI=">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</latexit>

W

T

=

1

Y

T
=
0
Y

Brady Neal

Increasing Data Efficiency

17 / 35TARNet inefficiency<latexit sha1_base64="m4is/x1qKs6D+47izt0tLlAKSRI=">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</latexit>

W

T

=

1

Y

T
=
0

Uses all the data

Brady NealY

Increasing Data Efficiency

17 / 35TARNet inefficiency<latexit sha1_base64="m4is/x1qKs6D+47izt0tLlAKSRI=">AAAWIHicrVhbbxtFFJ62AUq4tUV5gZelTqVW2hjbtASpslQpSYUQRa2U3oijai9j78h76+44brLyT+BP8D945w3xCL+DH8A5Z2ZvvjuqLe/OnjnnO/fZGduxL1LZav1z5eq1rQ8+/Oj6x9uffPrZ51/cuHnrRRqNEoc/dyI/Sl7ZVsp9EfLnUkifv4oTbgW2z1/awwOcf3nGk1RE4bE8j/lpYA1C0ReOJYH05ua133o2H4gwk2J4EQtHjhI+2Tb0547xS+TytHjuhfBo3D1uRyMZj+Q9w5LG3ZbZumecuIk1NqUnnKEZiFAEo8AYC1d63eYDJyhIHhcDT3Y7TnBqZLs9gHGigO9OHs6oaOUqTmzuR+NuuwlCRtQ3cuWn712nJ1yXh6DR531pdCu6zPPUE33Z3Ws3v0eYy2ie1SdC5WChThlwWcecKOxHo9CFbOfOVRJ5xzhyB9VUxpb0Chs4zBUBeFgR02x5aIgPrT+x7OiMmwb3zXcqNs22tkNC/cmAh+CT0d6dVMplLWBK90rgFgG3ZoDXh91rdh5sYPH6gVgAPM/iHg/dWue9udFoNVv0MWYHbT1oMP15Gt28/gfrMZdFzGEjFjDOQiZh7DOLpfA9YW3WYjHQTlkGtARGguY5m7BtkB0BFwcOC6hDuA7g6URTQ3hGzJSkHdDiwy8BSYPd0TwujPtEVXfUb1R4F+nICBttPIe7rTEDoErmAXWVXM65rhz6JMHCH8gXAXbGREEvnZpHfbj78CzBfryeAyeHkQtSCYwcoPlAVRTUkcBdxRU99yjOFvFxGKFNi72xwfbFmcD5CL5DwLJgnJKlaKvBHuuoh6SZk63I41PGFiO+Aw+Vfcuw+oUPgrKqvEDeY6AM2QWM6sjLdFZrZzGX1MiTObqQXxJHCNwpUCKqawHxE8BRr0QHMFGHRZkckC8x1UdTI/9E83mtxpDPPbInpawZVE2C9MS6b0q7czt9uNuEnYB8BnMe6cFYmGAPJ+yEajOPokncCTyN6ckhG50pepP6DvNpgk9YXSbgRjAjCiwVCRXP3CJlXQY0Q3/32M+kn0PeTYoK1rUJshFZqKo4JV2hzn0XVg2MSwBXpGJkc4pJOgKyw6NqGQKt1KeQUN+34EVK/eNr1AzuPkUtJhSTtGN8xjQOyCfM6Ih0j2Hskh6UNmAla7J9bcekphVjI2jtmdVqUP1h76p1b1oryoXUqYauyi5parH72mPMANpdjZqj1zq14qbk3WTKKpvsL1cNrNG6fShV5qeeK5eq0SOpOi5Gcb6f83zpUJ+YlEuM/QA4uiTV1/SUejPWmrYr/XeguySifDh0D3RNqH5HbeOpmQzogY6UD3yqz/GNgzNvAdOimupR9KrzyxCPtHy+1tvwzYi6XO4FdVZdNoFRVswslxc04hQfNcKMK7yYYh2TL98UP6Mysy72+8ELZxCxYqVGWhfdpdqY7/kxa8Ba0ICqqscTM42UhPYeMYx3C85dwEXeIWkJgYIdnhXzk5U5OATKREesT3N5H5Y2CFr5XOLs6cpTHKqj5BQHdvVyrViX8+pG0ZfLukW912UVfbksUs6oNoVeD1LCerVCTlIeUCKoVOnxCqmI3gmKrmx+vYZ9Ib1rR7ouEm3hyxWSAa0xyqtId+WTlfZJWgUTHROU+fUSETyj9W9CO6BN41jKyo2iWcqdXyqmpfx4w8iWksGG8S0lL1ZIcno3iIKGMs9W9hRyPV2J/LbYl6guMulqQ3TwXezqFcfQ7xHMtnrrdam3Tf0Oy99i+e4zraxvgrSXO9OMdhFjvRLm+4k90qR8+xF66Qm8o3F8wL6D/UEbro9rq+a6qLhPH+l+q+J2AHkf1rtDWhc3x431bnqvtueu6zgCDflvok8nJbev95CLkDbROx9Z1bo7s5ed1qR2Kx5wqR38+QrEsnvm4y3P73xkdY6OaAdZ7uPfB3L1jLIeXlkzi0+Uq865i09fMdz7cPVpRUg17yFVnU8rA2fP9QlRvX0PZs66WL8d6Aqs3ez2rSl9dfzp09OATj8j4Ch51C5YkrSiVREXncNikpF6t5HSmaU8mTVrJy2FN21bWqmrQON19WrF9aq0/eZGoz3938zs4EWn2X7QbD3rNB7d1//bXGdfs9vsLsRpnz2CznwKcXWu/bf11VZja3fn950/d/7a+VuxXr2iZb5ktc/Ov/8Dnqa3Pw==</latexit>

Only uses treatedgroup data

W

T

=

1

Y

T
=
0
Y

Brady Neal

Increasing Data Efficiency

17 / 35TARNet inefficiency<latexit sha1_base64="m4is/x1qKs6D+47izt0tLlAKSRI=">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</latexit>

Only uses treatedgroup data

W

T

=

1

Y

T
=
0
Y

Only uses
control group data
Brady Neal

Increasing Data Efficiency

17 / 35

X-Learner(Künzel et al., 2019)

Brady Neal

Increasing Data Efficiency

18 / 35

X-Learner(Künzel et al., 2019)

1. Estimate µ̂1 (x) and µ̂0 (x)<latexit sha1_base64="osuKu1R2XKoXiNSjjtTsB55nyHo=">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</latexit>

Brady Neal<latexit sha1_base64="kf88cwsMZn/C6e41SHA8x7p9rqo=">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</latexit>

Increasing Data Efficiency

18 / 35

X-Learner(Künzel et al., 2019)

1. Estimate µ̂1 (x) and µ̂0 (x)<latexit sha1_base64="osuKu1R2XKoXiNSjjtTsB55nyHo=">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</latexit>

Brady Neal<latexit sha1_base64="kf88cwsMZn/C6e41SHA8x7p9rqo=">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</latexit>

Assume X is a sufficient adjustment
set and is all observed covariates

Increasing Data Efficiency

18 / 35

X-Learner(Künzel et al., 2019)

1. Estimate µ̂1 (x) and µ̂0 (x)<latexit sha1_base64="osuKu1R2XKoXiNSjjtTsB55nyHo=">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</latexit><latexit sha1_base64="kf88cwsMZn/C6e41SHA8x7p9rqo=">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</latexit>

Assume X is a sufficient adjustment
set and is all observed covariates

2a. Impute ITEs

Brady Neal

Increasing Data Efficiency

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X-Learner(Künzel et al., 2019)

1. Estimate µ̂1 (x) and µ̂0 (x)<latexit sha1_base64="osuKu1R2XKoXiNSjjtTsB55nyHo=">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</latexit><latexit sha1_base64="kf88cwsMZn/C6e41SHA8x7p9rqo=">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</latexit>

2a. Impute ITEs

Assume X is a sufficient adjustment
set and is all observed covariates

Treatment group:<latexit sha1_base64="PFp+vEFb+AGb+R5v0HjXiL4voUE=">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</latexit>⌧ˆ1,i = Yi (1)Brady Nealµ̂0 (xi )Increasing Data Efficiency

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X-Learner(Künzel et al., 2019)

1. Estimate µ̂1 (x) and µ̂0 (x)<latexit sha1_base64="osuKu1R2XKoXiNSjjtTsB55nyHo=">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</latexit><latexit sha1_base64="kf88cwsMZn/C6e41SHA8x7p9rqo=">AAASrXicrVjbbttGEN2kN9e9OepjXtQoBVJAVmQ3hfNiIICdoCgawAF8SRslBkVREiFSZEjKqkPooV/T1/YD+iH9m545uxRJ3R3EgsjV7MyZ2bntrtuh58ZJs/nfrdsfffzJp59tfb79xZdfff3Nzp3KeRyMIts5swMviF62rdjx3KFzlriJ57wMI8fy255z0R4cyfzFlRPFbjA8Ta5D57Vv9YZu17WtBKTLnbutvpWkLX80uWw+aNnBlRW5VuLEV5b3w+VOrdlo8q86P9gzg5oyfyfBna1/VUt1VKBsNVK+ctRQJRh7ylIxPq/UnmqqELTXKgUtwsjlvKMmahuyI3A54LBAHeDZw69XhjrEb8GMKW1Di4dvBMmq+t7wdDDukqrfor9a4F2mIyW22HiNd9tg+qAmqg/qOrmMc1M5WVMCCx9zLS7sDEmRVdqlFXXx9vA7gf3yvAang1EHUhFGNmgeqJoiOiK8tV9l5X362SKfg5HYtHw1bdi+PBIyH+AzAJaFcUxLxdaqema8PqRmh7YKj8eILUf8AyvU9q3C6k7X4DKqehXCewrKQL3DqIy8Smcxd5ZzJQZ5skCX8CfkGII7BiVgXrvwnwuOcibawBQdFiPZ41pC5kfDIP/C+SxXQ8Rzl/bEjFqV2eRST2jqJrc7s9PDu03sCPIp5vrUI76owx6H2BFzM/NindwRfo35y6aN9gy9wbqTeNaxJsmuOnADzLhTLO0J7c/MIm1dClrVfHbVr9TvIO51ekXyug7ZgBbqLI6pa2hif4iuIX7x8RSqeDaj1KnDpx19ZssAtFyfRhJ9D7GKmPXjGdQUb49eC4lSp3bxz5hjn2uSiI6oe4xxh3pEuopO1lAHxo5JSav4xmXvmddaZf5J7eq+N6tV5Ias1KrJykNqaqpHZsUSAbG76DXb9DrdcWOubjJjVZv2511DcrRsn0jl8SnHqsNs7FOqjCteXLzORWvZZ53UGUvxfQ8ch5TqGnrM2gyNpu1C/R2ZKgkYD5tv3+SErnfRNp6ZSUH3jac88Ok6lx1HZt4C02JOtei94vwqxKdGPuv1bXxSUlfLnbOyyrIRRul0ZrW8y5FD/+iRRFzjhfR1yLV8N/1WCzObYn8YvOEcomRsYpA2Re8wNxav/FTV0AtqyKqyPyXSQol49ggxvj/lvA9c4R1QyxAUqfB0Oj9ZG4NjUCbGY13OZXWY2+Cy83XI2TKZpzl0RSUzHFLVq7VKXi7KG01fLduZ5ntZVtNXywrlirnpmn4QE+vlGrmEcRAJv5Clp2ukAu4Jmq5t/m0D+4bca0cmLyJj4cUaSZ89Rq8qMFX5fK19CbtgZHwiMr+/hwev2P8mPAHd1I+5bHIjb+Zy1+/l01x+fEPP5pL+Df2bS75bI+lwb3CnNJF5sbamhOtkLfLb6blEV1Gdzza8I3txx3ScqtlHJNp61ztkbdfNHpbtYtnpMy70N5fa85NpylPE2HTC7DyxS016bT+jlp5jj5bxkfoR54M9PJ+VuuamqHJOH5l6K+LuA/kA/e6YffHmuKE5Te+WztxlHU+hIftOzO0k5/bMGXIZ0k30LkbWud6ZO8vOatKnlT649An+eg1iXj2L8VbHdzGyvkcHPEHm5/gPgVy8o2yGl+fM8hvlunvu8ttXiHcXT48dITa8x8w6j53BUWfmhqh336O5u67k7z6qQnI3vVeZ0VfGn7099Xj7GYEj59Gn4ITSmlZEXHYPCymTmNNGzDtLfjNrlG5aGm/WtriQV77BOzTdyjFdaftyp7Y3+7+Z+cH5fmPvp0bzxaPak8fm/zZb6q66px7ATwfqCSrzBH611Z/qL/W3+qfysHJWaVXeaNbbt4zMt6r0V+n9Dw8cnJ4=</latexit>

2a. Impute ITEs

Assume X is a sufficient adjustment
set and is all observed covariates

Control group:

Treatment group:<latexit sha1_base64="PFp+vEFb+AGb+R5v0HjXiL4voUE=">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</latexit><latexit sha1_base64="e2TR8WssXFJDPNOPL2kEvLBGOwo=">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</latexit>⌧ˆ1,i = Yi (1)Brady Nealµ̂0 (xi )⌧ˆ0,i = µ̂1 (xi )Increasing Data Efficiency

Yi (0)

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X-Learner(Künzel et al., 2019)

1. Estimate µ̂1 (x) and µ̂0 (x)<latexit sha1_base64="osuKu1R2XKoXiNSjjtTsB55nyHo=">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</latexit><latexit sha1_base64="kf88cwsMZn/C6e41SHA8x7p9rqo=">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</latexit>

2a. Impute ITEs

Assume X is a sufficient adjustment
set and is all observed covariates

Control group:

Treatment group:<latexit sha1_base64="PFp+vEFb+AGb+R5v0HjXiL4voUE=">AAAS03icrVjbbttGEN2kN9e9OepjX9goLRyAViU3hfMiIIWdoCgawAF8SRsFAkVR1kK8haSsOoReir72O/oV/YS+tujfdObsUiR1oxTEgsjV7MyZ2bntrnuhK+Ok2fzv1u133n3v/Q92Ptz96ONPPv1s707tIg7Gke2c24EbRM97Vuy40nfOE5m4zvMwciyv5zqXvdExz19eO1EsA/8suQmdl5515cuBtK2ESN297ztDK0k7iTWedtOWKafG122jE4wTO/Ccrtxv3TcODMXjEUtzv2MH11YkrcSJry23K+939+rNRhN/xuKgpQd1of9Ogzs7f4mO6ItA2GIsPOEIXyQ0doUlYvq8EC3RFCHRXoqUaBGNJOYdMRW7JDsmLoc4LKKO6HlFv15oqk+/GTOGtE1aXPpGJGmIrzRPn8YDUNWb9RsF3lU6UmCzjTf07mlMj6iJGBK1Si7j3FSO15SQhQ+xFkl2hqDwKu3Sigb0dul3Qvbz84Y4HRr1SSqikU00l6iKwjoieiu/8sqH8LMFPodGbNPq1fTI9tWR4PmAPiPCsmgcw1K21RBPtNd9aHZgK/O4iNhqxF9phcq+dViD2RokoqpWwbxnRBmJ1zQqI6/TWcyd1VyJRp4u0cX8CTh84o6JEiCvJflPEkc5E23CZB0WInmFtYTIj4ZG/hHzWa6GFM8D2BMjagaySUJPqOsmtzuz06V3D9gRyac0N4Qe9oVJ9jjAjpCbmRdNcEf0a4JfNmy05+gN1B3H06Q1cXaZhBvQjJxhKU8of2YWKetSohn6cyB+gn6H4m7CK5zXJskGsFBlcQxdvo59m7oG+8WjJ1PZsxnFhA4PdgyRLSOi5foUEuv7hlYRo35cjZrS24XXQqCY0M7+mWDsYU0c0TF0T2jchx6WNqiTNcSRtmNa0sq+keg9i1oN5B/Xrup781pZzkelGjor29DUFA/0ijkCbHfRa7budarjxljddM6qHuzPuwbnaNk+lsrjU45VH9k4hFQZl724fJ3L1nKIOjERS/b9FXG0ITXQ9Bi1GWpNu4X6O9ZVEiAeNt6ezglV76xtMjeTEt3TnnKJT9U57zg884owLeRUB94rzq9DfKzls17fo08K6nq5C1RWWTaiUTqbWS8vMXLgHzXiiCu8EL4OsZYvZ1+jMLMp9tvB8xcQOWMTjbQpeh+5sXzlZ6JOvaBOWVX2J0eaKRHOHiGN78047xEu846gxScKV3g6m59WxuCEKFPtsQHmsjrMbZDofH1wdnTmKQ5VUckcB1f1eq2cl8vyRtHXy/Zn+V6WVfT1sky5Rm5K3Q9iYD2vkEsQB5bwCll6ViEVYE9QdGXzzxvY52OvHeu8iLSFlxWSHnqMWlWgq/JppX0JumCkfcIyv7yBB6/R/6Y4AW3rx1w22cqbudzNG/k0l59s6dlc0tvSv7nk6wpJB3uDnNFY5lllTTHXaSXyq9m5RFWRiWePvMN7cV93HEPvIxxtteu1Udum3sOyXSw7fcaF/iahPT+ZpjhFTHQnzM4TB9Ck1vYD1dJT2qN5fCy+pfNBi55PSl1zU1Q+p491vRVxDwn5iPrdCfri9rihPk0flM7cZR2PSUP2nerbSc7t6jPkKqRt9C5HVrneXzjLzmtSp5UhcakT/E0FYl49y/HWx3c5srpHBzhB5uf4t4FcvKNshpfnzOobZdU9d/XtK6T3gJ4uOkKseU+QdS46gyPO9Q1R7b7HC3ddzt9DqgrO3fRubU5fGX/+9nSF28+YOHIedQpOIK1oRcRV97AQMok+bcS4s+Q3s0bppqXw5m2LC3nlaby27laO7kq73b16a/5/M4uDi8NG67tG89mD+qOH+v82O+ILcVfsk5+OxCOqzFPyqy3+FH+Lf8S/tfNaWvut9rtivX1Ly3wuSn+1P/4HOQ+q6A==</latexit><latexit sha1_base64="e2TR8WssXFJDPNOPL2kEvLBGOwo=">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</latexit>⌧ˆ1,i = Yi (1)µ̂0 (xi )⌧ˆ0,i = µ̂1 (xi )Yi (0)

2b. Fit a model ⌧ˆ1 (x) to predict ⌧ˆ1,i from xi in treatment group<latexit 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Brady Neal

Increasing Data Efficiency

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X-Learner(Künzel et al., 2019)

1. Estimate µ̂1 (x) and µ̂0 (x)<latexit sha1_base64="osuKu1R2XKoXiNSjjtTsB55nyHo=">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</latexit><latexit sha1_base64="kf88cwsMZn/C6e41SHA8x7p9rqo=">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</latexit>

2a. Impute ITEs

Assume X is a sufficient adjustment
set and is all observed covariates

Control group:

Treatment group:<latexit sha1_base64="PFp+vEFb+AGb+R5v0HjXiL4voUE=">AAAS03icrVjbbttGEN2kN9e9OepjX9goLRyAViU3hfMiIIWdoCgawAF8SRsFAkVR1kK8haSsOoReir72O/oV/YS+tujfdObsUiR1oxTEgsjV7MyZ2bntrnuhK+Ok2fzv1u133n3v/Q92Ptz96ONPPv1s707tIg7Gke2c24EbRM97Vuy40nfOE5m4zvMwciyv5zqXvdExz19eO1EsA/8suQmdl5515cuBtK2ESN297ztDK0k7iTWedtOWKafG122jE4wTO/Ccrtxv3TcODMXjEUtzv2MH11YkrcSJry23K+939+rNRhN/xuKgpQd1of9Ogzs7f4mO6ItA2GIsPOEIXyQ0doUlYvq8EC3RFCHRXoqUaBGNJOYdMRW7JDsmLoc4LKKO6HlFv15oqk+/GTOGtE1aXPpGJGmIrzRPn8YDUNWb9RsF3lU6UmCzjTf07mlMj6iJGBK1Si7j3FSO15SQhQ+xFkl2hqDwKu3Sigb0dul3Qvbz84Y4HRr1SSqikU00l6iKwjoieiu/8sqH8LMFPodGbNPq1fTI9tWR4PmAPiPCsmgcw1K21RBPtNd9aHZgK/O4iNhqxF9phcq+dViD2RokoqpWwbxnRBmJ1zQqI6/TWcyd1VyJRp4u0cX8CTh84o6JEiCvJflPEkc5E23CZB0WInmFtYTIj4ZG/hHzWa6GFM8D2BMjagaySUJPqOsmtzuz06V3D9gRyac0N4Qe9oVJ9jjAjpCbmRdNcEf0a4JfNmy05+gN1B3H06Q1cXaZhBvQjJxhKU8of2YWKetSohn6cyB+gn6H4m7CK5zXJskGsFBlcQxdvo59m7oG+8WjJ1PZsxnFhA4PdgyRLSOi5foUEuv7hlYRo35cjZrS24XXQqCY0M7+mWDsYU0c0TF0T2jchx6WNqiTNcSRtmNa0sq+keg9i1oN5B/Xrup781pZzkelGjor29DUFA/0ijkCbHfRa7budarjxljddM6qHuzPuwbnaNk+lsrjU45VH9k4hFQZl724fJ3L1nKIOjERS/b9FXG0ITXQ9Bi1GWpNu4X6O9ZVEiAeNt6ezglV76xtMjeTEt3TnnKJT9U57zg884owLeRUB94rzq9DfKzls17fo08K6nq5C1RWWTaiUTqbWS8vMXLgHzXiiCu8EL4OsZYvZ1+jMLMp9tvB8xcQOWMTjbQpeh+5sXzlZ6JOvaBOWVX2J0eaKRHOHiGN78047xEu846gxScKV3g6m59WxuCEKFPtsQHmsjrMbZDofH1wdnTmKQ5VUckcB1f1eq2cl8vyRtHXy/Zn+V6WVfT1sky5Rm5K3Q9iYD2vkEsQB5bwCll6ViEVYE9QdGXzzxvY52OvHeu8iLSFlxWSHnqMWlWgq/JppX0JumCkfcIyv7yBB6/R/6Y4AW3rx1w22cqbudzNG/k0l59s6dlc0tvSv7nk6wpJB3uDnNFY5lllTTHXaSXyq9m5RFWRiWePvMN7cV93HEPvIxxtteu1Udum3sOyXSw7fcaF/iahPT+ZpjhFTHQnzM4TB9Ck1vYD1dJT2qN5fCy+pfNBi55PSl1zU1Q+p491vRVxDwn5iPrdCfri9rihPk0flM7cZR2PSUP2nerbSc7t6jPkKqRt9C5HVrneXzjLzmtSp5UhcakT/E0FYl49y/HWx3c5srpHBzhB5uf4t4FcvKNshpfnzOobZdU9d/XtK6T3gJ4uOkKseU+QdS46gyPO9Q1R7b7HC3ddzt9DqgrO3fRubU5fGX/+9nSF28+YOHIedQpOIK1oRcRV97AQMok+bcS4s+Q3s0bppqXw5m2LC3nlaby27laO7kq73b16a/5/M4uDi8NG67tG89mD+qOH+v82O+ILcVfsk5+OxCOqzFPyqy3+FH+Lf8S/tfNaWvut9rtivX1Ly3wuSn+1P/4HOQ+q6A==</latexit><latexit sha1_base64="e2TR8WssXFJDPNOPL2kEvLBGOwo=">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</latexit>⌧ˆ1,i = Yi (1)µ̂0 (xi )⌧ˆ0,i = µ̂1 (xi )Yi (0)

2b. Fit a model ⌧ˆ1 (x) to predict ⌧ˆ1,i from xi in treatment group
Fit a model ⌧ˆ0 (x) to predict ⌧ˆ0,i from xi in control group<latexit 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Brady Neal

Increasing Data Efficiency

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X-Learner(Künzel et al., 2019)

1. Estimate µ̂1 (x) and µ̂0 (x)<latexit sha1_base64="osuKu1R2XKoXiNSjjtTsB55nyHo=">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</latexit><latexit sha1_base64="kf88cwsMZn/C6e41SHA8x7p9rqo=">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</latexit>

2a. Impute ITEs

Assume X is a sufficient adjustment
set and is all observed covariates

Control group:

Treatment group:<latexit sha1_base64="PFp+vEFb+AGb+R5v0HjXiL4voUE=">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</latexit><latexit sha1_base64="e2TR8WssXFJDPNOPL2kEvLBGOwo=">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</latexit>⌧ˆ1,i = Yi (1)µ̂0 (xi )⌧ˆ0,i = µ̂1 (xi )Yi (0)

2b. Fit a model ⌧ˆ1 (x) to predict ⌧ˆ1,i from xi in treatment group
Fit a model ⌧ˆ0 (x) to predict ⌧ˆ0,i from xi in control group<latexit 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3.

<latexit sha1_base64="EMunY3MlgWATYCP6VALasYQMI4g=">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</latexit>⌧ˆ(x) = g(x) ⌧ˆ0 (x) + (1 g(x)) ⌧ˆ1 (x)where g(x) is some weighing function between 0 and 1. Example: propensity score<latexit sha1_base64="yWtQkPwj1Y5gecocBX7yqmiIGQI=">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</latexit>

Brady Neal

Increasing Data Efficiency

18 / 35

Question:
What would motivate someone to consider
a more complex type of estimation than
COM/GCOM?

Conditional Outcome Modeling
Increasing Data Efficiency
Propensity Scores and IPW
Other Methods

Brady Neal

Propensity Scores and IPW

20 / 35

Propensity scores

Brady Neal

Propensity Scores and IPW

21 / 35

Propensity scorese(W ) , P (T = 1 | W )<latexit sha1_base64="oFsTp51xydUEyPGVLoEpiS4jiV8=">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</latexit>

Brady Neal

Propensity Scores and IPW

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Propensity scorese(W ) , P (T = 1 | W )<latexit sha1_base64="oFsTp51xydUEyPGVLoEpiS4jiV8=">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</latexit><latexit sha1_base64="oFsTp51xydUEyPGVLoEpiS4jiV8=">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</latexit>

Brady Neal

Propensity Scores and IPW

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Propensity scorese(W ) , P (T = 1 | W )<latexit sha1_base64="oFsTp51xydUEyPGVLoEpiS4jiV8=">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</latexit><latexit sha1_base64="oFsTp51xydUEyPGVLoEpiS4jiV8=">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</latexit>

Given positivity, unconfoundedness given W implies unconfoundedness
given the propensity score e(W ).
<latexit sha1_base64="y6nnSigj2UkfW7tvC9oZQnFjROI=">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</latexit><latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit>

Brady Neal

Propensity Scores and IPW

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Propensity scorese(W ) , P (T = 1 | W )<latexit sha1_base64="oFsTp51xydUEyPGVLoEpiS4jiV8=">AAAS+3icrVhLb9tGEN6kL9d9OeqxFzZKAReQFclN4VwEpLATFEUDOIAfaaPAoMiVRIivkJRVh9Cv6aFA0Wt/Rg+9tOi/6cy3S5HUi1IQCyJXszPfzM5rd90LXSdOWq3/bt1+59333v9g58Pdjz7+5NPP9u7ULuJgHFny3ArcIHreM2PpOr48T5zElc/DSJpez5WXvdExz19eyyh2Av8suQnlS88c+E7fscyESFd733XDeL9rBX4/GPs2MX5tdJPIMf2BK18Z3XCffkkz8aSfGB2jbXQ9xzZK/Fd79VazhT9jcdDWg7rQf6fBnZ2/RVfYIhCWGAtPSOGLhMauMEVMnxeiLVoiJNpLkRItopGDeSmmYpdkx8QlicMk6oieA/r1QlN9+s2YMaQt0uLSNyJJQ3yleWwa90FVb9ZvFHhX6UiBzTbe0LunMT2iJmJI1Cq5jHNTOV5TQhY+xFocsjMEhVdplVbUp7dLvxOyn583xClpZJNURCOLaC5RFYV1RPRWfuWVD+FnE3ySRmzT6tX0yPbVkeD5gD4jwjJpHMNSttUQT7TXfWiWsJV5XERsNeIvtEJl3zqs/mwNDqKqVsG8Z0QZidc0KiOv01nMndVciUaeLtHF/Ak4fOKOiRIgrx3yn0Mc5Uy0CJN1mIjkAGsJkR9NjfwD5rNcDSmeB7AnRtQMZJMDPaGum9zuzE6X3j1gRySf0twQetgXDbJHAjtCbmZebIA7ol8T/LJgozVHb6LuOJ4NWhNnV4NwA5pxZljKE8qfmUXKupRohv4ciB+hX1LcG/AK53WDZANYqLI4hi5fx75DXYP94tGTqezZjNKADg92DJEtI6Ll+hQS67tPq4hRP65GTentwmshUBrQzv6ZYOxhTRzRMXRPaGxDD0sb1Mma4kjbMS1pZd846D2LWg3kH9eu6nvzWlnOR6UaOis70NQSD/SKOQJsd9Frlu51quPGWN10zqoe7M+7Budo2T6WyuNTjpWNbBxCqozLXly+zmVrOUSdNBBL9v2AODqQ6mt6jNoMtabdQv0d6yoJEA8Lb0/nhKp31jaZm0mJ7mlPucSn6px3HJ55RZgmcqoL7xXn1yE+1vJZr+/RJwV1vdwFKqssG9Eonc2sl3cwkvCPGnHEFV4IX4dYy5ezr1GY2RT77eD5C4icsYlG2hTdRm4sX/mZqFMvqFNWlf3JkWZKhLNHSON7M857hMu8I2jxicIVns7mp5UxOCHKVHusj7msDnMbHHQ+G5xdnXmKQ1VUMsfBVb1eK+flsrxR9PWy9izfy7KKvl6WKdfITUf3gxhYzyvkEsSBJbxClp5VSAXYExRd2fzTBvb52GvHOi8ibeFlhaSHHqNWFeiqfFppX4IuGGmfsMzPb+DBa/S/KU5A2/oxl0228mYud/NGPs3lJ1t6Npf0tvRvLvm6QlJib3BmNJZ5VllTzHVayaXyqWpPkLSbZOcXVW0NPHvkRd6zbd2ZDL3fcFao3bGDHtDQe12222Wn1LjQBx1oz0+wKU4bE90xs3PHATQpH3xPNfeU9nIeH4tv6BzRpueTUnfdFJXP82Ndl0XcQ0I+or54gv65PW6oT90HpbN5Wcdj0pB9p/oWk3O7+qy5CmkbvcuRVU3YC2feeU3qVDMkLnXSv6lAzKtsOd76+C5HVvftACfN/Lz/NpCLd5nN8PKcWX3zrLoPr76lhfTu09NF54g17wmyzkUHkeJc3yTVLn28cCfm/D2kquDcTe/W5vSV8edvWQPcksbEkfOo03ICaUUrIq66r4WQSfSpJMbdJr/BNUs3MoU3b1tcyCtP43V0t5K6K+1e7dXb8//DWRxcHDbb3zZbzx7UHz3U/9/ZEV+Iu2Kf/HQkHlFlnpJfLfGb+Ev8I/6tTWu/1n6v/aFYb9/SMp+L0l/tz/8Bqte0Yg==</latexit><latexit sha1_base64="oFsTp51xydUEyPGVLoEpiS4jiV8=">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</latexit>

Given positivity, unconfoundedness given W implies unconfoundedness
given the propensity score e(W ).
<latexit 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Even if W is high-dimensional, e(W ) is only 1-dimensional!
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Brady Neal

Propensity Scores and IPW

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Propensity Score Theorem
Given positivity, unconfoundedness given W implies unconfoundedness
given the propensity score e(W ).
<latexit sha1_base64="y6nnSigj2UkfW7tvC9oZQnFjROI=">AAASkXicrVjbbttGEN0kvbjuzVEe+8JGKdAHWZHdFE4eDBi1ExRtA9iAb20cBBJFWYR4C0lZdQR9QV/bz+mH9G965uxSJHVXEAsiV7MzZ2bntrtuRZ6bpI3Gf3fu3vvo408+3fhs8/Mvvvzq6637lfMk7Me2c2aHXhhftpqJ47mBc5a6qedcRrHT9Fuec9HqHcr8xY0TJ24YnKa3kfPab14Hbse1mylIJxdvtqqNeoN/1vRgxwyqyvwdh/c3/lVXqq1CZau+8pWjApVi7KmmSvB5pXZUQ0WgvVZD0GKMXM47aqQ2IdsHlwOOJqg9PK/x65WhBvgtmAmlbWjx8I0haanvDE8b4w6p+i36rQLvPB1DYouNt3i3DKYPaqq6oC6TyzhXlZM1pbDwKdfiws6IFFmlXVpRB28Pv1PYL89bcDoYtSEVY2SD5oGqKaIjxlv7VVbepZ+b5HMwEpvmr6YF2+dHQuZDfHrAamKc0FKx1VIvjNcDanZoq/B4jNh8xD+xQm3fIqzOeA0uo6pXIbynoPTUO4zKyIt0FnNnPldqkEczdAl/So4A3AkoIfPahf9ccJQz0Qam6GgyktdcS8T8qBvkXzif5WqEeG7TnoRRs5hNLvVEpm5yuzM7PbxbxI4hP8Rcl3rEFzXY4xA7Zm5mXqyRO8avAX/ZtNGeoNdZdxLPGtYk2VUDbogZd4ylPaH9mVmkrRuCZpnPtvqN+h3EvUavSF7XIBvSQp3FCXUFJvb76BriFx9PoYpnM0qNOnza0WW29EDL9Wkk0fcYq0hYP55BHeLt0WsRUWrULv4ZcOxzTRLRPnUPMG5Tj0hb6GR1tWfsGJW0im9c9p5prRbzT2pX971JrSIXsFItk5X71NRQT8yKJQJid9Frtul1uuMmXN1owqoW7c+7huRo2T6RyuNTjlWb2dilVBlXvDh7nbPWsss6qTGW4vtrcOxTqmPoCWszMpo2C/V3aKokZDxsvn2TE7reRdtgYmYIum885YFP17nsODLzFphN5tQVvVecX4T43Mhnvb6Fz5DUxXLnrKyybIzRcDyzWN7lyKF/9EgirvEi+jriWr4df63CzKrYHwYvmEKUjE0N0qrobebG7JWfqip6QRVZVfanRFooMc8eEcaPxpyPgCu8PWoJQJEKH47nR0tjcATKyHisw7msDnMbXHa+NjmvTOZpDl1R6QSHVPVirZKXs/JG0xfLtsf5XpbV9MWyQrlhbrqmHyTEulwilzIOIuEXsvR0iVTIPUHTtc2/r2BfwL22b/IiNhZeLJH02WP0qkJTlS+X2peyC8bGJyLzx3t48Ib9b8QT0Lp+zGXTtbyZy92+l09z+cGans0l/TX9m0u+WyLpcG9wxzSROVlaU8J1vBT57fhcoquoxmcL3pG9uG06jmX2EYm23vX2Wds1s4dlu1h2+kwK/c2l9vxkOuQpYmA6YXae2KYmvbafUUsvsUfL+FD9gPPBDp4vSl1zVVQ5p/dNvRVxd4G8h353xL64Pm5kTtPbpTN3WcdzaMi+I3M7ybk9c4ach7SO3tnIOtfbU2fZSU36tNIFlz7B3y5BzKtnNt7i+M5G1vfokCfI/Bz/IZCLd5TV8PKcmX+jXHbPnX/7ivDu4OmxIySG94hZ57EzOOrM3BD17ns4ddeV/N1FVUjuDh9WJvSV8SdvT9e8/fTBkfPoU3BKaU0rIs67h0WUSc1pI+GdJb+Z1Us3LY03aVtSyCvf4O2bbuWYrrT5Zqu6M/m/menB+W5958d64+RJ9eCp+b/NhvpGPVTfw0976gCVeQy/yvn3L/W3+qfyoPKsclD5SbPevWNkHqjSX+XX/wHwV5FB</latexit><latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit>

Brady Neal

Propensity Scores and IPW

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Propensity Score Theorem
Given positivity, unconfoundedness given W implies unconfoundedness
given the propensity score e(W ). Equivalently,<latexit sha1_base64="y6nnSigj2UkfW7tvC9oZQnFjROI=">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</latexit><latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit>(Y (1), Y (0)) ?? T | W =) (Y (1), Y (0)) ?? T | e(W )<latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit>

Brady Neal

Propensity Scores and IPW

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Propensity Score Theorem
Given positivity, unconfoundedness given W implies unconfoundedness
given the propensity score e(W ). Equivalently,<latexit sha1_base64="y6nnSigj2UkfW7tvC9oZQnFjROI=">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</latexit><latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit>(Y (1), Y (0)) ?? T | W =) (Y (1), Y (0)) ?? T | e(W )<latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit><latexit sha1_base64="+nLdlJQ0Pu/fpBBeP9g62yMnaeY=">AAAVDnicrVhLb9tGEN6oL9d9xalvvbCxAyQArcpuihQIBARwEhRFUySo82gsI6DIlUSIr5KrqA6h/9Bf00OBotdee+yt1/ZXdObbpUjqaQWRYHE5O/PNY2dmd91NAj9Trdbflxpvvf3Ou+9tvb/9wYcfffzJ5Z0rT7J4lLrysRsHcfqs62Qy8CP5WPkqkM+SVDphN5BPu8Njnn/6UqaZH0cn6jyRZ6HTj/ye7zqKSC92Gj90urLvR7nyh68S31WjVE62LfO5Zn0fezKbvnciej3NlKPkmXW948ZRLx5FHuHfsBxlXW/ZrRtWvl+d2J/cXiRue6kzbkdxBKCE5E+7MojHVjvuWVX5MyvvJAMnUnFIwElW17o/WQzPoIrioEIZqTp2wpD75ewSAxkhHik3DiXJp35/oKz2kRtajDEVBpRhK4Aq0bvn9SvRu9ZJHDWYjZskHh2C2/OcSclQulOxdwXgpvyFs/PclUAu5O3IyKvlz4vLe61mCx9rfnBoBnvCfB7GO1t/iY7wRCxcMRKhkCISisaBcERG31NxKFoiIdqZyImW0sjHvBQTsU2yI+KSxOEQdUi/fXo7NdSI3hkzg7RLWgL6S0nSEtcMj0fjHqj6yfqtCu8yHTmw2cZzenYNZkhUJQZEXSdXcF5Ujn1SZOHX8MUnOxNQ2Eu35lGPngG9K7Kff8+JU9LII6mURi7RAqJqCutI6anjyp4PEGcHfJJGbNNyb7pk+/KV4PmYvkPCcmicwVK21RL3TdQjaJawlXkCrNhyxJ/JQ23fKqze1Acfq6q9YN4TogzFKxrVkVfprObOci5lkCcLdDG/AkdE3BlRYuS1T/HziaOeiS5hsg4HK9mHLwnyo2mQv8V8kasJrecB7MmwahayyYeexNRNaXdhZ0DPLrBTks9pbgA9HAub7JHATpGbRRRtcKf0NsabCxvdGXoTdcfraZNPnF024cY040+xdCR0PAuLtHU50SzzPRDfQb+kdbcRFc5rm2RjWKizOIOuyKx9m7oGxyWkX6ZyZAuKDR0h7BggW4ZEK/VpJNb3BXmRoX4Cg5rTM0DUEqDY0M7xGWMcwide0RF0j2nsQQ9LW9TJmuKWsWNS08qx8dF75rVayD+uXd33ZrWyXIRKtUxWtqGpJW4aj3kF2O5q1FzT63THzeDdZMaqLuwvuwbnaN0+lirXp75WHrJxAKk6LkdxsZ+LfDlCndhYS459nzjakOoZeobaTIym7Ur9HZsqibEeLp6hyQld76xtPDOTEz00kQqIT9c57zg88xNhOsipDqJXnV+FeM/IF72+S98c1NVyT1BZddmURvl0ZrW8j5FEfPSIV1zjJYh1Al8+n/5ZlZmLYr8ZvGgOkTNWGaSLonvIjcWen4g96gV7lFX1ePJKMyXF2SOh8f6Uc59wmXcILRFRuMLz6fxk7RrcJcrERKyHuaIOSxt8dD4PnB2TeZpDV5Sa4eCqXq2V83JR3mj6allvmu91WU1fLcuUl8hN3/SDDFjP1sgprANLhJUsPVkjFWNP0HRt848XsC/CXjsyeZEaC5+ukQzRY7RXsanKB2vtU+iCqYkJyzx/jQi+RP+b4AS0aRxLWbVRNEu589eKaSk/3jCypWS4YXxLyVdrJCX2Bn9KY5lHa2uKuR6u5dL5tG5PkLSbFOcXXW02frsURd6zPdOZLLPfcFbo3bGNHmCbva7Y7YpTalbpgz60lyfYHKeNsemYxbnjAJp0DL6hmntAezmPj8WXdI44pN/7te56UVQ+z49MXVZxjwj5FvXFu+ifm+Mm5tR9UDub13XcIw3F38TcYkruwJw1lyFtoncxsq4Jb+7MO6tJn2oGxKVP+udrEMsqW4y3en0XI+v7doyTZnnefxPI1bvMxfDKnFl+81x3H15+S0vo2aPfAJ0jM7x3kXUBOogUj81NUu/Sx3N3Ys7fI6oKzt386pUZfXX82VtWH7ekEXGUPPq0rCCtaVXEZfe1BDLKnEoy3G3KG1yzdiPTeLO2ZZW8Cg1e23QrabrS9ovLe4ez/8OZHzw5ah5+1Ww9Otq7c9P8f2dLfCauiusUp1viDlXmQ4qr2/it8U/j38Z/u7/s/rr7++4fmrVxych8Kmqf3T//B3KvWf0=</latexit>

Graphical Proof:W

T

Brady NealY

Propensity Scores and IPW

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Propensity Score Theorem
Given positivity, unconfoundedness given W implies unconfoundedness
given the propensity score e(W ). Equivalently,<latexit sha1_base64="y6nnSigj2UkfW7tvC9oZQnFjROI=">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</latexit><latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit>(Y (1), Y (0)) ?? T | W =) (Y (1), Y (0)) ?? T | e(W )<latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit><latexit sha1_base64="+nLdlJQ0Pu/fpBBeP9g62yMnaeY=">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</latexit>

W

Graphical Proof:

P (T | W )<latexit sha1_base64="RknOQawkM/4W5EV5APGqYeOoRPY=">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</latexit>

T

Brady NealY

Propensity Scores and IPW

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Propensity Score Theorem
Given positivity, unconfoundedness given W implies unconfoundedness
given the propensity score e(W ). Equivalently,<latexit sha1_base64="y6nnSigj2UkfW7tvC9oZQnFjROI=">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</latexit><latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit>(Y (1), Y (0)) ?? T | W =) (Y (1), Y (0)) ?? T | e(W )<latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">AAATJ3icrVjbbttGEF2nN9e9xFEf+8JEKWC3siq7KZwXAwHsBEXRAA7gWxsFhkStLMK8haTsOoI+ox/Rj+g39K1oH/sdfemZsyuRlHWxg0gQuZydOTM7N+6qHftemjUa/yzdee/9Dz78aPnjlU8+/ezzu6v3Kkdp1E9cfehGfpSctFup9r1QH2Ze5uuTONGtoO3r4/b5rswfX+gk9aLwILuK9augdRZ6Xc9tZSCdrv621oz6mRsFem1zveaMHxrr607TCzs61riEmdPMAJsFHAZex2m6UdiN+phMUnAGMFanztui6bUi3vrparVRb/DjXB9s2kFV2c9+dG/5D9VUHRUpV/VVoLQKVYaxr1oqxfel2lQNFYP2Sg1ASzDyOK/VUK1Atg8uDY4WqOe4nuHppaWGeBbMlNIutPj4JZB01FeWp4Nxl1RzF/1OgXeWjgGxxcYr3NsWMwA1Uz1QF8mNOG8qJ2vKYOFjrsWDnTEpskq3tKIu7j6eM9gv1ytwaow6kEowckHzQTUU0ZHgbvwqK+/Rzy3yaYzEptmracP22ZGQ+Qjfc2C1ME5pqdjqqGfW6yE1a9oqPD4jNhvxV6zQ2DcPqzteg8eomlUI7wEo5+oNRmXkeTqLuTObK7PIwym6hD8jRwjuFJSIee3Bfx44ypnoAlN0tBjJM64lZn7ULfKPnB/laox4btCelFFzmE0e9cS2bnK7R3b6uLeJnUB+gLke9YgvarBHEzthbo68WCN3gqdLPrm00Z2g11l3Es8a1iTZVQNuhBlvjGU8Yfw5sshYNwDNsd8N9RP1a8S9Rq9IXtcgG9FCk8UpdYU29jvoGuKXAFehimdHlBp1BLSjx2w5By3XZ5BE37dYRcr68S3qAHefXouJUqN28c8lxwHXJBHtU/clxh3qEWkHnayutq0dw5JW8Y3H3nNdq8P8k9o1fW9Sq8iFrFTHZuUONTXUI7tiiYDYXfSaa3ud6bgpVzecsKpN+/OuITlatk+k8viUY9VhNvYoVcYVL05f57S1bLFOaoyl+P4MHDuU6lp6ytqMraaVQv3t2iqJGA+X98DmhKl30XY5MTMAPbCe8sFn6lzeODLzGpgt5lST3ivOz0N8auVHvb6N74DU+XJHrKyybILRYDwzX97jSNM/ZiQRN3gxfR1zLffHP6cwc1Psd4MXXkOUjM0s0k3RO8yN6Ss/UFX0giqyquxPibRQEu49YowfjjkfAld4z6klBEUqfDCeHy6MwR4oQ+uxLudGdZjb4LHzdcjZtJlnOExFZRMcUtXztUpeTssbQ58v2xnne1nW0OfLCuWCuenZfpAS62SBXMY4iERQyNKDBVIR3wmGbmz++Qb2hXzX9m1eJNbC4wWSAXuMWVVkq/L5QvsydsHE+kRkfnkLD16w/w25A7qtH3PZ7FbezOWu3sqnufzlLT2bSwa39G8u+WaBpOa7wRvTRObFwpoSrv2FyK/H+xJTRTVe2/COvIs7tuM49j0i0TZvvR3Wds2+w0ZvsdHuMy30N4/a853pgLuIS9sJR/uJDWoya/sBtfQc72gZ76rvsD/YxPVZqWveFFX26X1bb0XcLSBvo9/tsS/eHje2u+mN0p67rOMpNIx+Q3s6ybl9u4echXQbvdORTa53ru1lJzWZ3UoPXGYHf7UAMa+e6Xjz4zsd2ZyjI+4g8338u0AunlFuhpfnzOwT5aJz7uzTV4x7F1efHSG1vHvMOp+dQatDe0I0b9/da2ddyd8tVIXk7uBBZUJfGX/y9HTG008fHDmP2QVnlDa0IuKsc1hMmczuNlKeWfKTWb100jJ4k7alhbwKLN6O7VbadqWV09Xq5uR/M9cHR1v1ze/rjRePqk8e2/9tltWX6oFag5+21RNU5j786qr/lu4vfb30TeX3yp+Vvyp/G9Y7S1bmC1X6VP79HwlRyS8=</latexit><latexit sha1_base64="+nLdlJQ0Pu/fpBBeP9g62yMnaeY=">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</latexit>

W

Graphical Proof:

P (T = 1 | W )<latexit sha1_base64="HKOiwODvc4MQlTpe9cVjLda9EJo=">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</latexit>

T

Brady NealY

Propensity Scores and IPW

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Propensity Score Theorem
Given positivity, unconfoundedness given W implies unconfoundedness
given the propensity score e(W ). Equivalently,<latexit sha1_base64="y6nnSigj2UkfW7tvC9oZQnFjROI=">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</latexit><latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit>(Y (1), Y (0)) ?? T | W =) (Y (1), Y (0)) ?? T | e(W )<latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit><latexit sha1_base64="+nLdlJQ0Pu/fpBBeP9g62yMnaeY=">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</latexit>

W

Graphical Proof:<latexit sha1_base64="K1oMagBWgBWi4J9/qCI4hSIC3MQ=">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</latexit>

e(W )

T

Brady NealY

Propensity Scores and IPW

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Propensity Score Theorem
Given positivity, unconfoundedness given W implies unconfoundedness
given the propensity score e(W ). Equivalently,<latexit sha1_base64="y6nnSigj2UkfW7tvC9oZQnFjROI=">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</latexit><latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit>(Y (1), Y (0)) ?? T | W =) (Y (1), Y (0)) ?? T | e(W )<latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit><latexit sha1_base64="UopTeG8ASlwcISOBmlfrReRpz+I=">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</latexit>

Graphical Proof:

W

e(W )

T

Brady NealY

Propensity Scores and IPW

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Propensity Score Theorem
Given positivity, unconfoundedness given W implies unconfoundedness
given the propensity score e(W ). Equivalently,<latexit sha1_base64="y6nnSigj2UkfW7tvC9oZQnFjROI=">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</latexit><latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit>(Y (1), Y (0)) ?? T | W =) (Y (1), Y (0)) ?? T | e(W )<latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit><latexit sha1_base64="UPh9SvP40VYuMCNm0FqUqLNhHZA=">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</latexit>

Graphical Proof:

W

e(W )

T

Brady NealY

Propensity Scores and IPW

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Propensity Score Theorem
Given positivity, unconfoundedness given W implies unconfoundedness
given the propensity score e(W ). Equivalently,<latexit sha1_base64="y6nnSigj2UkfW7tvC9oZQnFjROI=">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</latexit><latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit>(Y (1), Y (0)) ?? T | W =) (Y (1), Y (0)) ?? T | e(W )<latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">AAATJ3icrVjbbttGEF2nN9e9xFEf+8JEKWC3siq7KZwXAwHsBEXRAA7gWxsFhkStLMK8haTsOoI+ox/Rj+g39K1oH/sdfemZsyuRlHWxg0gQuZydOTM7N+6qHftemjUa/yzdee/9Dz78aPnjlU8+/ezzu6v3Kkdp1E9cfehGfpSctFup9r1QH2Ze5uuTONGtoO3r4/b5rswfX+gk9aLwILuK9augdRZ6Xc9tZSCdrv621oz6mRsFem1zveaMHxrr607TCzs61riEmdPMAJsFHAZex2m6UdiN+phMUnAGMFanztui6bUi3vrparVRb/DjXB9s2kFV2c9+dG/5D9VUHRUpV/VVoLQKVYaxr1oqxfel2lQNFYP2Sg1ASzDyOK/VUK1Atg8uDY4WqOe4nuHppaWGeBbMlNIutPj4JZB01FeWp4Nxl1RzF/1OgXeWjgGxxcYr3NsWMwA1Uz1QF8mNOG8qJ2vKYOFjrsWDnTEpskq3tKIu7j6eM9gv1ytwaow6kEowckHzQTUU0ZHgbvwqK+/Rzy3yaYzEptmracP22ZGQ+Qjfc2C1ME5pqdjqqGfW6yE1a9oqPD4jNhvxV6zQ2DcPqzteg8eomlUI7wEo5+oNRmXkeTqLuTObK7PIwym6hD8jRwjuFJSIee3Bfx44ypnoAlN0tBjJM64lZn7ULfKPnB/laox4btCelFFzmE0e9cS2bnK7R3b6uLeJnUB+gLke9YgvarBHEzthbo68WCN3gqdLPrm00Z2g11l3Es8a1iTZVQNuhBlvjGU8Yfw5sshYNwDNsd8N9RP1a8S9Rq9IXtcgG9FCk8UpdYU29jvoGuKXAFehimdHlBp1BLSjx2w5By3XZ5BE37dYRcr68S3qAHefXouJUqN28c8lxwHXJBHtU/clxh3qEWkHnayutq0dw5JW8Y3H3nNdq8P8k9o1fW9Sq8iFrFTHZuUONTXUI7tiiYDYXfSaa3ud6bgpVzecsKpN+/OuITlatk+k8viUY9VhNvYoVcYVL05f57S1bLFOaoyl+P4MHDuU6lp6ytqMraaVQv3t2iqJGA+X98DmhKl30XY5MTMAPbCe8sFn6lzeODLzGpgt5lST3ivOz0N8auVHvb6N74DU+XJHrKyybILRYDwzX97jSNM/ZiQRN3gxfR1zLffHP6cwc1Psd4MXXkOUjM0s0k3RO8yN6Ss/UFX0giqyquxPibRQEu49YowfjjkfAld4z6klBEUqfDCeHy6MwR4oQ+uxLudGdZjb4LHzdcjZtJlnOExFZRMcUtXztUpeTssbQ58v2xnne1nW0OfLCuWCuenZfpAS62SBXMY4iERQyNKDBVIR3wmGbmz++Qb2hXzX9m1eJNbC4wWSAXuMWVVkq/L5QvsydsHE+kRkfnkLD16w/w25A7qtH3PZ7FbezOWu3sqnufzlLT2bSwa39G8u+WaBpOa7wRvTRObFwpoSrv2FyK/H+xJTRTVe2/COvIs7tuM49j0i0TZvvR3Wds2+w0ZvsdHuMy30N4/a853pgLuIS9sJR/uJDWoya/sBtfQc72gZ76rvsD/YxPVZqWveFFX26X1bb0XcLSBvo9/tsS/eHje2u+mN0p67rOMpNIx+Q3s6ybl9u4echXQbvdORTa53ru1lJzWZ3UoPXGYHf7UAMa+e6Xjz4zsd2ZyjI+4g8338u0AunlFuhpfnzOwT5aJz7uzTV4x7F1efHSG1vHvMOp+dQatDe0I0b9/da2ddyd8tVIXk7uBBZUJfGX/y9HTG008fHDmP2QVnlDa0IuKsc1hMmczuNlKeWfKTWb100jJ4k7alhbwKLN6O7VbadqWV09Xq5uR/M9cHR1v1ze/rjRePqk8e2/9tltWX6oFag5+21RNU5j786qr/lu4vfb30TeX3yp+Vvyp/G9Y7S1bmC1X6VP79HwlRyS8=</latexit><latexit sha1_base64="ZwD937NUd3UMuUBHyuHRqe+s4wk=">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</latexit>

Graphical Proof:

W

e(W )

T

Brady NealY

Propensity Scores and IPW

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Propensity Score Theorem
Given positivity, unconfoundedness given W implies unconfoundedness
given the propensity score e(W ). Equivalently,<latexit sha1_base64="y6nnSigj2UkfW7tvC9oZQnFjROI=">AAASkXicrVjbbttGEN0kvbjuzVEe+8JGKdAHWZHdFE4eDBi1ExRtA9iAb20cBBJFWYR4C0lZdQR9QV/bz+mH9G965uxSJHVXEAsiV7MzZ2bntrtuRZ6bpI3Gf3fu3vvo408+3fhs8/Mvvvzq6637lfMk7Me2c2aHXhhftpqJ47mBc5a6qedcRrHT9Fuec9HqHcr8xY0TJ24YnKa3kfPab14Hbse1mylIJxdvtqqNeoN/1vRgxwyqyvwdh/c3/lVXqq1CZau+8pWjApVi7KmmSvB5pXZUQ0WgvVZD0GKMXM47aqQ2IdsHlwOOJqg9PK/x65WhBvgtmAmlbWjx8I0haanvDE8b4w6p+i36rQLvPB1DYouNt3i3DKYPaqq6oC6TyzhXlZM1pbDwKdfiws6IFFmlXVpRB28Pv1PYL89bcDoYtSEVY2SD5oGqKaIjxlv7VVbepZ+b5HMwEpvmr6YF2+dHQuZDfHrAamKc0FKx1VIvjNcDanZoq/B4jNh8xD+xQm3fIqzOeA0uo6pXIbynoPTUO4zKyIt0FnNnPldqkEczdAl/So4A3AkoIfPahf9ccJQz0Qam6GgyktdcS8T8qBvkXzif5WqEeG7TnoRRs5hNLvVEpm5yuzM7PbxbxI4hP8Rcl3rEFzXY4xA7Zm5mXqyRO8avAX/ZtNGeoNdZdxLPGtYk2VUDbogZd4ylPaH9mVmkrRuCZpnPtvqN+h3EvUavSF7XIBvSQp3FCXUFJvb76BriFx9PoYpnM0qNOnza0WW29EDL9Wkk0fcYq0hYP55BHeLt0WsRUWrULv4ZcOxzTRLRPnUPMG5Tj0hb6GR1tWfsGJW0im9c9p5prRbzT2pX971JrSIXsFItk5X71NRQT8yKJQJid9Frtul1uuMmXN1owqoW7c+7huRo2T6RyuNTjlWb2dilVBlXvDh7nbPWsss6qTGW4vtrcOxTqmPoCWszMpo2C/V3aKokZDxsvn2TE7reRdtgYmYIum885YFP17nsODLzFphN5tQVvVecX4T43Mhnvb6Fz5DUxXLnrKyybIzRcDyzWN7lyKF/9EgirvEi+jriWr4df63CzKrYHwYvmEKUjE0N0qrobebG7JWfqip6QRVZVfanRFooMc8eEcaPxpyPgCu8PWoJQJEKH47nR0tjcATKyHisw7msDnMbXHa+NjmvTOZpDl1R6QSHVPVirZKXs/JG0xfLtsf5XpbV9MWyQrlhbrqmHyTEulwilzIOIuEXsvR0iVTIPUHTtc2/r2BfwL22b/IiNhZeLJH02WP0qkJTlS+X2peyC8bGJyLzx3t48Ib9b8QT0Lp+zGXTtbyZy92+l09z+cGans0l/TX9m0u+WyLpcG9wxzSROVlaU8J1vBT57fhcoquoxmcL3pG9uG06jmX2EYm23vX2Wds1s4dlu1h2+kwK/c2l9vxkOuQpYmA6YXae2KYmvbafUUsvsUfL+FD9gPPBDp4vSl1zVVQ5p/dNvRVxd4G8h353xL64Pm5kTtPbpTN3WcdzaMi+I3M7ybk9c4ach7SO3tnIOtfbU2fZSU36tNIFlz7B3y5BzKtnNt7i+M5G1vfokCfI/Bz/IZCLd5TV8PKcmX+jXHbPnX/7ivDu4OmxIySG94hZ57EzOOrM3BD17ns4ddeV/N1FVUjuDh9WJvSV8SdvT9e8/fTBkfPoU3BKaU0rIs67h0WUSc1pI+GdJb+Z1Us3LY03aVtSyCvf4O2bbuWYrrT5Zqu6M/m/menB+W5958d64+RJ9eCp+b/NhvpGPVTfw0976gCVeQy/yvn3L/W3+qfyoPKsclD5SbPevWNkHqjSX+XX/wHwV5FB</latexit><latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit>(Y (1), Y (0)) ?? T | W =) (Y (1), Y (0)) ?? T | e(W )<latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit><latexit sha1_base64="ZwD937NUd3UMuUBHyuHRqe+s4wk=">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</latexit>

Graphical Proof:

W

e(W )

T

Brady Neal

See non-graphical proof in
Appendix A.2 of the course bookY

Propensity Scores and IPW

22 / 35

Implications for the
Positivity-Unconfoundedness Tradeoff

Brady Neal

Propensity Scores and IPW

23 / 35

Implications for the
Positivity-Unconfoundedness Tradeoff
Recall that overlap decreases with thedimensionality of the adjustment set

Brady Neal

Propensity Scores and IPW

23 / 35

Implications for the
Positivity-Unconfoundedness Tradeoff
Recall that overlap decreases with thedimensionality of the adjustment set

W<latexit sha1_base64="VPvP5OxvTxlPGVcROXV6+AtJi24=">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</latexit>

Brady Neal

Propensity Scores and IPW

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Implications for the
Positivity-Unconfoundedness Tradeoff
Recall that overlap decreases with thedimensionality of the adjustment set
The propensity score magicallyreduces the dimensionality of theadjustment set done to 1!

Brady Neal<latexit sha1_base64="yHnJqHzzwl4G8ppVn8qDn+vJ3Kk=">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</latexit>e(W )Propensity Scores and IPW

W<latexit sha1_base64="VPvP5OxvTxlPGVcROXV6+AtJi24=">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</latexit>

23 / 35

Implications for the
Positivity-Unconfoundedness Tradeoff
Recall that overlap decreases with thedimensionality of the adjustment set
The propensity score magicallyreduces the dimensionality of theadjustment set done to 1!

<latexit sha1_base64="yHnJqHzzwl4G8ppVn8qDn+vJ3Kk=">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</latexit>

e(W )

W
<latexit sha1_base64="VPvP5OxvTxlPGVcROXV6+AtJi24=">AAASkXicrVjbbttGEN2kN9e9OcpjX9goBfogq5Kbwu2DAaN2gqJtABvwrY2DQKIoixBvISmrjqAv6Gv7Of2Q/k3PnF2KpO4KYkHkanbmzOzcdtftyHOTtNH47979997/4MOPtj7e/uTTzz7/YudB5SIJB7HtnNuhF8ZX7VbieG7gnKdu6jlXUey0/LbnXLb7RzJ/eevEiRsGZ+ld5Lz0WzeB23XtVgrS6eWrnWqj3uCfNTtomkFVmb+T8MHWv+padVSobDVQvnJUoFKMPdVSCT4vVFM1VATaSzUCLcbI5byjxmobsgNwOeBogdrH8wa/XhhqgN+CmVDahhYP3xiSlvra8HQw7pKq36LfKvAu0jEitth4h3fbYPqgpqoH6iq5jHNdOVlTCgt/4Fpc2BmRIqu0Syvq4u3hdwr75XkHTgejDqRijGzQPFA1RXTEeGu/ysp79HOLfA5GYtPi1bRh++JIyHyITx9YLYwTWiq2WuqZ8XpAzQ5tFR6PEVuM+CdWqO1bhtWdrMFlVPUqhPcMlL56g1EZeZnOYu4s5koN8niOLuFPyRGAOwElZF678J8LjnIm2sAUHS1G8oZriZgfdYP8C+ezXI0Qz13akzBqFrPJpZ7I1E1ud2anh3eb2DHkR5jrUY/4ogZ7HGLHzM3MizVyx/g15C+bNtpT9DrrTuJZw5oku2rADTHjTrC0J7Q/M4u0dSPQLPPZVb9Rv4O41+gVyesaZENaqLM4oa7AxP4AXUP84uMpVPFsRqlRh087esyWPmi5Po0k+r7FKhLWj2dQR3h79FpElBq1i3+GHPtck0R0QN1DjDvUI9IWOlld7Rs7xiWt4huXvWdWq8X8k9rVfW9aq8gFrFTLZOUBNTXUE7NiiYDYXfSabXqd7rgJVzeesqpN+/OuITlatk+k8viUY9VhNvYoVcYVL85f57y17LFOaoyl+P4GHAeU6hp6wtqMjKbtQv0dmSoJGQ+bb9/khK530TacmhmB7htPeeDTdS47jsy8BmaLOXVN7xXnlyE+NfJZr2/jMyJ1udwFK6ssG2M0mswsl3c5cugfPZKIa7yIvo64lq8mX6swsy72u8ELZhAlY1ODtC56h7kxf+VnqopeUEVWlf0pkRZKzLNHhPHjCedj4Apvn1oCUKTCR5P58coYHIMyNh7rci6rw9wGl52vQ85rk3maQ1dUOsUhVb1cq+TlvLzR9OWynUm+l2U1fbmsUG6Zm67pBwmxrlbIpYyDSPiFLD1bIRVyT9B0bfPva9gXcK8dmLyIjYWXKyR99hi9qtBU5fOV9qXsgrHxicj88RYevGX/G/MEtKkfc9l0I2/mcndv5dNcfrihZ3NJf0P/5pJvVkg63BvcCU1kTlfWlHCdrER+PTmX6Cqq8dmGd2Qv7piOY5l9RKKtd70D1nbN7GHZLpadPpNCf3OpPT+ZjniKGJpOmJ0ndqlJr+1n1NJz7NEyPlLf4XzQxPNZqWuuiyrn9IGptyLuHpD30e+O2Rc3x43MaXq3dOYu63gKDdl3bG4nObdnzpCLkDbROx9Z53pn5iw7rUmfVnrg0if4uxWIefXMx1se3/nI+h4d8gSZn+PfBXLxjrIeXp4zi2+Uq+65i29fEd5dPD12hMTwHjPrPHYGR52bG6LefY9m7rqSv3uoCsnd0aPKlL4y/vTt6Ya3nwE4ch59Ck4prWlFxEX3sIgyqTltJLyz5DezeummpfGmbUsKeeUbvAPTrRzTlbZf7VSb0/+bmR1c7NWb39cbp0+qh4fm/zZb6kv1SH0DP+2rQ1TmCfwq59+/1N/qn8rDyo+Vw8pPmvX+PSPzUJX+Kr/+D/K/kUk=</latexit>

Unfortunately, we don’t have accessto it. The best we can do is model it,
shifting the high-dimensionality
problem to the modeling of e(W ) , P (T = 1 | W )<latexit sha1_base64="jLupkGKl6KgsQ3x2L+BioPxjjiM=">AAAS8HicrVhLb9tGEN6kL9d9OeqxFzZKAQeQVclN61wEBLATFEUDOIAfaaPAoMiVRIivkJRVh9D/6KFA0Wt/Rq/tpcf+m858uxRJvSgFsSByNTvzzey8dte90HXipNX679btd9597/0Pdj7c/ejjTz79bO9O7SIOxpElz63ADaLnPTOWruPL88RJXPk8jKTp9Vx52Rsd8/zltYxiJ/DPkptQvvTMge/0HctMiHS19103jPcv7xvdJHJMf+DKV0Y33Kdf0kw86SdGx2gbXc+xja4V+P1g7NuEdv9qr95qtvBnLA7aelAX+u80uLPzr+gKWwTCEmPhCSl8kdDYFaaI6fNCtEVLhER7KVKiRTRyMC/FVOyS7Ji4JHGYRB3Rc0C/XmiqT78ZM4a0RVpc+kYkaYivNI9N4z6o6s36jQLvKh0psNnGG3r3NKZH1EQMiVoll3FuKsdrSsjCh1iLQ3aGoPAqrdKK+vR26XdC9vPzhjgljWySimhkEc0lqqKwjojeyq+88iH8bIJP0ohtWr2aHtm+OhI8H9BnRFgmjWNYyrYa4on2ug/NErYyj4uIrUb8hVao7FuH1Z+twUFU1SqY94woI/GaRmXkdTqLubOaK9HI0yW6mD8Bh0/cMVEC5LVD/nOIo5yJFmGyDhORHGAtIfKjqZF/wHyWqyHF8wD2xIiagWxyoCfUdZPbndnp0rsH7IjkU5obQg/7okH2SGBHyM3Miw1wR/Rrgl8WbLTm6E3UHcezQWvi7GoQbkAzzgxLeUL5M7NIWZcSzdCfA/Ej9EuKewNe4bxukGwAC1UWx9Dl69h3qGuwXzx6MpU9m1Ea0OHBjiGyZUS0XJ9CYn1f0ypi1I+rUVN6u/BaCJQGtLN/Jhh7WBNHdAzdExrb0MPSBnWypjjSdkxLWtk3DnrPolYD+ce1q/revFaW81Gphs7KDjS1xAO9Yo4A2130mqV7neq4MVY3nbOqB/vzrsE5WraPpfL4lGNlIxuHkCrjsheXr3PZWg5RJw3Ekn0/II4OpPqaHqM2Q61pt1B/x7pKAsTDwtvTOaHqnbVN5mZSonvaUy7xqTrnHYdnXhGmiZzqwnvF+XWIj7V81ut79ElBXS93gcoqy0Y0Smcz6+UdjCT8o0YccYUXwtch1vLl7GsUZjbFfjt4/gIiZ2yikTZFt5Eby1d+JurUC+qUVWV/cqSZEuHsEdL43ozzHuEy7whafKJwhaez+WllDE6IMtUe62Muq8PcBgedzwZnV2ee4lAVlcxxcFWv18p5uSxvFH29rD3L97Ksoq+XZco1ctPR/SAG1vMKuQRxYAmvkKVnFVIB9gRFVzb/tIF9Pvbasc6LSFt4WSHpoceoVQW6Kp9W2pegC0baJyzz8xt48Br9b4oT0LZ+zGWTrbyZy928kU9z+cmWns0lvS39m0u+rpCU2BucGY1lnlXWFHOdVnKpfKraEyTtJtn5RVVbA88eeZH3bFt3JkPvN5wVanfsoAc09F6X7XbZKTUu9EEH2vMTbIrTxkR3zOzccQBNygffU809pb2cx8fiGzpHtOn5pNRdN0Xl8/xY12UR95CQj6gvnqB/bo8b6lP3QelsXtbxmDRk36m+xeTcrj5rrkLaRu9yZFUT9sKZd16TOtUMiUud9G8qEPMqW463Pr7LkdV9O8BJMz/vvw3k4l1mM7w8Z1bfPKvuw6tvaSG9+/R00TlizXuCrHPRQaQ41zdJtUsfL9yJOX8PqSo4d9O7tTl9Zfz5W9YAt6QxceQ86rScQFrRioir7mshZBJ9Kolxt8lvcM3SjUzhzdsWF/LK03gd3a2k7kq7V3v19vz/cBYHF4fN9rfN1rMH9UcP9f93dsQX4q7YJz8diUdUmafkV0v8Jv4Sf4t/alHt19rvtT8U6+1bWuZzUfqr/fk/rqOvSQ==</latexit>

Brady Neal

Propensity Scores and IPW

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Questions:
1. What is the intuition behind why we cancondition on e(W) instead of W?
2. What is attractive about conditioning one(W) as opposed to W?
3. Why does this not solve positivity issues
when W is high-dimensional?

Pseudo-populations

Brady Neal

Propensity Scores and IPW

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Pseudo-populations
Regular population<latexit sha1_base64="x9MwVMOEAp+s2GhuQc6v5IjE2Eg=">AAAUb3icrVjpbttGEF67l+tedmKgPwIUbOwANiCrstvCAQIDKWwHRdoUCeA4aS3DoMiVRIhXyVVUh9Bb9UX6s4/RN+jMt0uR1O0gEiwuZ2e+OXZmdtet2PdS1Wj8u7L6wYcfffzJ2qfrn33+xZdfbWzeuUijfuLIl07kR8nrlp1K3wvlS+UpX76OE2kHLV++avVOeP7VG5mkXhSeq5tYXgV2J/TanmMrIl1vrvzdbMmOF2bK672NPUf1Ezlct8zngfVb5Mp09N4M6fUyVbaSV9Zu04nCdtQPXcLfs2xl7TZqjT0r2ylP7AwfzRJXZKgKZKj2rMuW9KOB5cu2so6jtlVGuGLEEe8cvKivnCiQI7TE63RnwBnWHKzk8JnbKTsc26o77qkklor5j5biz82b5C4FYipvU4ZuZYGuN7Yb9QY+1uTgwAy2hfk8jzbX/hFN4YpIOKIvAiFFKBSNfWGLlL6X4kA0REy0K5ERLaGRh3kphmKdZPvEJYnDJmqPfjv0dmmoIb0zZgpph7T49JeQpCUeGB6Xxm1Q9ZP1WyXeWToyYLONN/RsGcyAqEp0ibpILudcVo59UmThQ/jikZ0xKOylU/GoTU+f3hXZz783xClp5JJUQiOHaD5RNYV1JPTUcWXPu4izDT5JI7Zptjctsn32SvB8RN8eYdk0TmEp22qJJybqITRL2Mo8PlZsNuJf5KG2bx5We+SDh1XVXjDvOVF64i2NqsjzdJZzZzaXMsjDKbqYX4EjJO6UKBHy2qP4ecRRzUSHMFmHjZXswJcY+VE3yE8xn+dqTOu5D3tSrJqFbPKgJzZ1U9id2+nTswXshOQzmutCD8eiRvZIYCfIzTyKNXAn9DbAmwMbnTF6HXXH61kjnzi7aoQb0Yw3wtKR0PHMLdLWZUSzzHdf/Ar9kta9hqhwXtdINoKFOotT6ArN2h9T1+C4BPTLVI5sTqlBRwA7usiWHtEKfRqJ9X1HXqSoH9+gZvT0EbUYKDVo5/gMMA7gE69oH7oHNHahh6Ut6mR1cWTsGFa0cmw89J5JrRbyj2tX971xrSwXolItk5XH0NQQPxiPeQXY7nLUHNPrdMdN4d1wzKoW7C+6Budo1T6WKtanulYusrELqSouR3G6n9N8OUSd1LCWHPsOcRxDqm3oKWozNprWS/V3Yqokwno4eAYmJ3S9s7bB2ExG9MBEyic+Xee84/DMn4RpI6eaiF55fh7imZHPe32Lvhmo8+UuUFlV2YRG2WhmvryHkUR89IhXXOPFiHUMX74d/VmlmWWx3w9eOIHIGasM0rLoLnJjuufnYpt6wTZlVTWevNJMSXD2iGm8M+LcIVzm7UFLSBSu8Gw0P1y4BqdEGZqItTGX12Fhg4fO54KzaTJPc+iKUmMcXNXztXJeTssbTZ8v647yvSqr6fNlmfIGuemZfpAC6/UCOYV1YImglKXnC6Qi7Amarm3+fQn7Quy1fZMXibHw1QLJAD1GexWZqny20D6FLpiYmLDMH+8QwTfof0OcgG4bx0JW3SqahdzNO8W0kB/cMrKFZHDL+BaSbxdISuwN3ojGMi8W1hRzPV/IpfNp0Z4gaTfJzy+62mr4bVEUec92TWeyzH7DWaF3x2P0gJrZ6/LdLj+lpqU+6EF7cYLNcNoYmI6Znzv2oUnH4GequWe0l/P4RHxP54gD+n1S6a7LovJ5vm/qsox7SMhH1BdP0T9vjxubU/d+5Wxe1XFGGvK/obnFFNy+OWvOQrqN3unIuibciTPvuCZ9qukSlz7p3yxALKpsOt789Z2OrO/bEU6axXn/fSCX7zLL4RU5M/vmueg+PPuWFtOzTb8+OkdqeE+RdT46iBQvzU1S79InE3dizt9DqgrO3ez+nTF9VfzxW1YHt6Q+cRQ8+rSsIK1pZcRZ97UYMsqcSlLcbYobXL1yI9N447alpbwKDN6x6VbSdKX1643tg/H/4UwOLg7rBz/WGy8Otx8/NP/fWRP3xH2xS3E6Eo+pMp9TXJ3Ve6s/rT5d/eXuf1tfb32zZWnW1RUjc1dUPlt7/wPCxyRj</latexit>

W

T

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Propensity Scores and IPW

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Pseudo-populations
Regular population<latexit sha1_base64="x9MwVMOEAp+s2GhuQc6v5IjE2Eg=">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</latexit>W

TY
P (T | W ) 6= P (T )<latexit sha1_base64="nXPPx5E5upvT2MNCbVd0JFQJD9c=">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</latexit>

Brady Neal

Propensity Scores and IPW

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Pseudo-populations
Regular population

Reweighted population(Pseudo-population)<latexit sha1_base64="x9MwVMOEAp+s2GhuQc6v5IjE2Eg=">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</latexit>W

TY
P (T | W ) 6= P (T )<latexit sha1_base64="nXPPx5E5upvT2MNCbVd0JFQJD9c=">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</latexit>

Brady Neal

P (T | W ) = P (T ) or<latexit sha1_base64="9kIxNPgC7uAfOrS/l3+QWHAZTDI=">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</latexit>

Propensity Scores and IPW

P (T | W ) = 1<latexit sha1_base64="NYa6JfkFwEVZRzboxfl8ts8u2dQ=">AAAS3nicrVhbbxtFFJ6WWwi31DzystRBKpJj7FCUvliqlLRCiEqplBvUVbDX63jlvbG7jkktv/KAhHjlZ/AK/4JH/g3f+WbW6/U9VWN5d3zmnO+cObeZSTvy3CSt1f67c/ett995972t97c/+PCjjz/ZuVc6S8JBbDunduiF8UW7lTieGzinqZt6zkUUOy2/7Tnn7f6hzJ9fO3HihsFJehM5L/3WVeB2XbuVgnS5s9uMHjRTSKS+E6RW03c7VtMOg244CDqQ+9JqWPXLnXKtWuOfNT+om0FZmb/j8N7Wv6qpOipUthooXzkqUCnGnmqpBJ8Xqq5qKgLtpRqBFmPkct5RY7UN2QG4HHC0QO3jeYVfLww1wG/BTChtQ4uHbwxJS31heDoYd0nVb9FvTfEu0zEitth4g3fbYPqgpqoH6jq5jHNTOVlTCgsfcS0u7IxIkVXahRV18fbwO4X98rwBp4NRB1IxRjZoHqiaIjpivLVfZeU9+rlFPgcjsWn5atqwfXkkZD7Epw+sFsYJLRVbLfXUeD2gZoe2Co/HiC1H/AUr1PatwupO1uAyqnoVwnsCSl+9wqiIvErndO4s50oN8niBLuFPyRGAOwElZF678J8LjmIm2sAUHS1G8opriZgfVYP8HeezXI0Qzz3akzBqFrPJpZ7I1E1ud2anh3eb2DHkR5jrUY/4ogJ7HGLHzM3MixVyx/g15C+bNtoz9CrrTuJZwZokuyrADTHjTrC0J7Q/M4u0dSPQLPPZU99Tv4O4V+gVyesKZENaqLM4oa7AxL6BriF+8fEUqng2o1Sow6cdPWZLH7Rcn0YSfV9hFQnrxzOoI7w9ei0iSoXaxT9Djn2uSSI6oO4hxh3qEWkLnayqDowd44JW8Y3L3jOv1WL+Se3qvjerVeQCVqplsrJBTTX10KxYIiB2T3vNNr1Od9yEqxvPWNWm/XnXkBwt2idSeXyKseowG3uUKuKKFxevc9Fa9lknFcZSfH8FjgaluoaesDYjo2l7qv4OTZWEjIfNt29yQte7aBvOzIxA942nPPDpOpcdR2Z+BmaLOdWk96bnVyE+MfJZr2/jMyJ1tdwZK6soG2M0msyslnc5cugfPZKIa7yIvo64ls8nX2tqZlPsN4MXzCFKxqYGaVP0DnNj8cpPVBm9oIysKvpTIi2UmGePCOPdCecucIW3Ty0BKFLho8n8eG0MjkAZG491OZfVYW6Dy87XIWfTZJ7m0BWVznBIVa/WKnm5KG80fbVsZ5LvRVlNXy0rlGvmpmv6QUKsizVyKeMgEv5Ulp6skQq5J2i6tvmHDewLuNcOTF7ExsLzNZI+e4xeVWiq8tla+1J2wdj4RGR+fA0PXrP/jXkCuq0fc9n0Vt7M5W5ey6e5/PCWns0l/Vv6N5d8tUbS4d7gTmgi83xtTQnX8VounU/r9gQHu0l2ftHVVuGzDS/Knt0xncky+41khd4dG+wBFbPXZbtddkpNpvqgS+35CXbE08bQdMzs3LFHTdoH36LmnmEvl/Gh+hrniDqeTwvddVNUOc8PTF1O4+4D+QB98Yj98/a4kTl17xXO5kUdT6Ah+47NLSbn9sxZcxnSbfQuRtY10Zk7885q0qeaHrj0Sf9mDWJeZYvxVsd3MbK+b4c8aebn/TeBPH2X2Qwvz5nlN8919+Hlt7QI7y6eHjtHYniPmHUeO4ijTs1NUu/Sh3N3YsnffVSF5O7ofmlGXxF/9pZ1xVvSABw5jz4tp5TWtGnEZfe1iDKpOZUkvNvkN7hq4Uam8WZtS6byyjd4DdOtHNOVti93yvXZ/+HMD872q/VvqrXnD8uPH5n/72ypz9R99QB+OlCPUZnH8KutflN/qb/VP6WfSr+Wfi/9oVnv3jEyn6rCX+nP/wGGU6hL</latexit>

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Pseudo-populations
Reweighted population(Pseudo-population)

Regular population<latexit sha1_base64="x9MwVMOEAp+s2GhuQc6v5IjE2Eg=">AAAUb3icrVjpbttGEF67l+tedmKgPwIUbOwANiCrstvCAQIDKWwHRdoUCeA4aS3DoMiVRIhXyVVUh9Bb9UX6s4/RN+jMt0uR1O0gEiwuZ2e+OXZmdtet2PdS1Wj8u7L6wYcfffzJ2qfrn33+xZdfbWzeuUijfuLIl07kR8nrlp1K3wvlS+UpX76OE2kHLV++avVOeP7VG5mkXhSeq5tYXgV2J/TanmMrIl1vrvzdbMmOF2bK672NPUf1Ezlct8zngfVb5Mp09N4M6fUyVbaSV9Zu04nCdtQPXcLfs2xl7TZqjT0r2ylP7AwfzRJXZKgKZKj2rMuW9KOB5cu2so6jtlVGuGLEEe8cvKivnCiQI7TE63RnwBnWHKzk8JnbKTsc26o77qkklor5j5biz82b5C4FYipvU4ZuZYGuN7Yb9QY+1uTgwAy2hfk8jzbX/hFN4YpIOKIvAiFFKBSNfWGLlL6X4kA0REy0K5ERLaGRh3kphmKdZPvEJYnDJmqPfjv0dmmoIb0zZgpph7T49JeQpCUeGB6Xxm1Q9ZP1WyXeWToyYLONN/RsGcyAqEp0ibpILudcVo59UmThQ/jikZ0xKOylU/GoTU+f3hXZz783xClp5JJUQiOHaD5RNYV1JPTUcWXPu4izDT5JI7Zptjctsn32SvB8RN8eYdk0TmEp22qJJybqITRL2Mo8PlZsNuJf5KG2bx5We+SDh1XVXjDvOVF64i2NqsjzdJZzZzaXMsjDKbqYX4EjJO6UKBHy2qP4ecRRzUSHMFmHjZXswJcY+VE3yE8xn+dqTOu5D3tSrJqFbPKgJzZ1U9id2+nTswXshOQzmutCD8eiRvZIYCfIzTyKNXAn9DbAmwMbnTF6HXXH61kjnzi7aoQb0Yw3wtKR0PHMLdLWZUSzzHdf/Ar9kta9hqhwXtdINoKFOotT6ArN2h9T1+C4BPTLVI5sTqlBRwA7usiWHtEKfRqJ9X1HXqSoH9+gZvT0EbUYKDVo5/gMMA7gE69oH7oHNHahh6Ut6mR1cWTsGFa0cmw89J5JrRbyj2tX971xrSwXolItk5XH0NQQPxiPeQXY7nLUHNPrdMdN4d1wzKoW7C+6Budo1T6WKtanulYusrELqSouR3G6n9N8OUSd1LCWHPsOcRxDqm3oKWozNprWS/V3Yqokwno4eAYmJ3S9s7bB2ExG9MBEyic+Xee84/DMn4RpI6eaiF55fh7imZHPe32Lvhmo8+UuUFlV2YRG2WhmvryHkUR89IhXXOPFiHUMX74d/VmlmWWx3w9eOIHIGasM0rLoLnJjuufnYpt6wTZlVTWevNJMSXD2iGm8M+LcIVzm7UFLSBSu8Gw0P1y4BqdEGZqItTGX12Fhg4fO54KzaTJPc+iKUmMcXNXztXJeTssbTZ8v647yvSqr6fNlmfIGuemZfpAC6/UCOYV1YImglKXnC6Qi7Amarm3+fQn7Quy1fZMXibHw1QLJAD1GexWZqny20D6FLpiYmLDMH+8QwTfof0OcgG4bx0JW3SqahdzNO8W0kB/cMrKFZHDL+BaSbxdISuwN3ojGMi8W1hRzPV/IpfNp0Z4gaTfJzy+62mr4bVEUec92TWeyzH7DWaF3x2P0gJrZ6/LdLj+lpqU+6EF7cYLNcNoYmI6Znzv2oUnH4GequWe0l/P4RHxP54gD+n1S6a7LovJ5vm/qsox7SMhH1BdP0T9vjxubU/d+5Wxe1XFGGvK/obnFFNy+OWvOQrqN3unIuibciTPvuCZ9qukSlz7p3yxALKpsOt789Z2OrO/bEU6axXn/fSCX7zLL4RU5M/vmueg+PPuWFtOzTb8+OkdqeE+RdT46iBQvzU1S79InE3dizt9DqgrO3ez+nTF9VfzxW1YHt6Q+cRQ8+rSsIK1pZcRZ97UYMsqcSlLcbYobXL1yI9N447alpbwKDN6x6VbSdKX1643tg/H/4UwOLg7rBz/WGy8Otx8/NP/fWRP3xH2xS3E6Eo+pMp9TXJ3Ve6s/rT5d/eXuf1tfb32zZWnW1RUjc1dUPlt7/wPCxyRj</latexit><latexit sha1_base64="CtPqyah4M0xn8axhaP4crWih7V8=">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</latexit>

W

W

T

Y
P (T | W ) 6= P (T )<latexit sha1_base64="nXPPx5E5upvT2MNCbVd0JFQJD9c=">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</latexit>

Brady NealTY
P (T | W ) = P (T ) or<latexit sha1_base64="9kIxNPgC7uAfOrS/l3+QWHAZTDI=">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</latexit>

Propensity Scores and IPW

P (T | W ) = 1<latexit sha1_base64="NYa6JfkFwEVZRzboxfl8ts8u2dQ=">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</latexit>

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Pseudo-populations
Reweighted population(Pseudo-population)

Regular population<latexit sha1_base64="x9MwVMOEAp+s2GhuQc6v5IjE2Eg=">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</latexit><latexit sha1_base64="CtPqyah4M0xn8axhaP4crWih7V8=">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</latexit>

W

W

T

Y

T

Y
P (T | W ) = P (T ) or

P (T | W ) 6= P (T )<latexit sha1_base64="nXPPx5E5upvT2MNCbVd0JFQJD9c=">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</latexit><latexit sha1_base64="9kIxNPgC7uAfOrS/l3+QWHAZTDI=">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</latexit>

P (T | W ) = 1<latexit sha1_base64="NYa6JfkFwEVZRzboxfl8ts8u2dQ=">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</latexit>

1
Reweighting intuition: P (T | W ) · P (T | W ) = 1<latexit sha1_base64="7bYfiGO9BLg1PVktDdKv2r6qy5s=">AAATD3icrVhLb9tGEN6kL9d9OeqxFzZOgRSQVclN4VwMBLATFEUDOIAfaaPAkEjKJsxXScqqQ+hH9Nf0UKDotccee+y1/RP95tulKOpFO4gFkavZmW9m57W77se+l2bt9t+3br/19jvvvrf2/voHH3708ScbdxrHaTRMbPfIjvwoed7vpa7vhe5R5mW++zxO3F7Q992T/sWezJ9cuknqReFhdhW7L4PeWegNPLuXgXS6cdSN73czSGSBG2ZWN/Acq2tH4SAahg7kvsQvJ8LEIOnZeWec1/GPrV2rc7qx2W61+WfNDzpmsKnM30F0Z+0v1VWOipSthipQrgpVhrGveirF54XqqLaKQXupctASjDzOu2qs1iE7BJcLjh6oF3ie4dcLQw3xWzBTStvQ4uObQNJSXxgeB+MBqfot+q0p3mU6cmKLjVd49w1mAGqmzkGtkys4rysna8pg4UOuxYOdMSmySruyogHePn5nsF+eV+B0MXIglWBkg+aDqimiI8Fb+1VWfk4/98jnYiQ2LV9NH7Yvj4TMR/hcAKuHcUpLxVZLPTFeD6nZpa3C4zNiyxF/xgq1fauwBpM1eIyqXoXwHoJyoV5hVEVepXM6d5ZzZQZ5vECX8GfkCMGdghIxrz34zwNHNRNtYIqOHiN5xrXEzI+WQf6O80WuxojnFu1JGTWL2eRRT2zqprS7sNPHu0/sBPI55s6pR3zRhD0usRPmZuHFJrkT/Brxl00b7Rl6i3Un8WxiTZJdTeBGmPEmWNoT2p+FRdq6HDTLfLbU99TvIu5NekXyugnZiBbqLE6pKzSx30XXEL8EeApVPFtQmtQR0I5zZssFaKU+jST6vsIqUtaPb1BzvH16LSZKk9rFPyOOA65JIjqk7hHGDvWItIVO1lI7xo5xRav4xmPvmddqMf+kdnXfm9UqciEr1TJZuUtNbfXArFgiIHZPe802vU533JSrG89Y1af9ZdeQHK3aJ1JlfKqxcpiN55Sq4ooXF69z0Vq2WSdNxlJ8fwaOXUoNDD1lbcZG0/pU/e2ZKokYD5vvwOSErnfRNpqZyUEPjKd88Ok6lx1HZn4CZo851aX3pudXIT428kWv7+OTk7pa7piVVZVNMMonM6vlPY5c+kePJOIaL6avY67l88nXmpq5LvabwQvnECVjM4N0XXSHubF45YdqE71gE1lV9adEWigJzx4xxvcmnPeAK7wX1BKCIhWeT+bHtTHYB2VsPDbgXFGHpQ0eO59Dzq7JPM2hKyqb4ZCqXq1V8nJR3mj6allnku9VWU1fLSuUS+amZ/pBSqznNXIZ4yASwVSWHtZIRdwTNF3b/MM17Au51w5NXiTGwpMayYA9Rq8qMlX5tNa+jF0wMT4RmR9fw4OX7H9jnoBu6sdSNruRN0u5q9fyaSk/uqFnS8nghv4tJV/VSLrcG7wJTWSe1daUcB3Ucul8qtsTXOwmxflFV1uTzz68KHu2YzqTZfYbyQq9O+6yBzTNXlfsdsUpNZ3qgx61lyfYnKeNkemYxblji5q0D75FzT3FXi7jPfU1zhEdPJ9Uuut1UeU8PzR1OY27DeQd9MV99s+b48bm1L1VOZtXdTyGhuI7NreYkts3Z81lSDfRuxhZ14Qzd+ad1aRPNefg0if9qxrEssoW462O72Jkfd+OeNIsz/tvAnn6LnM9vDJnlt886+7Dy29pMd4DPH12jtTw7jPrfHYQVx2Zm6Tepffm7sSSv9uoCsnd/G5jRl8Vf/aWdcZb0hAcJY8+LWeU1rRpxGX3tZgymTmVpLzblDe4VuVGpvFmbUun8ioweLumW7mmK62fbmx2Zv+HMz843m51vmm1nz3YfPTQ/H9nTX2m7qr78NOOeoTKPIBfbfWb+kf9q/5r/NL4tfF74w/NevuWkflUVf4af/4PvOm8uQ==</latexit>

Brady Neal

Propensity Scores and IPW

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Inverse probability weighting (IPW)

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Propensity Scores and IPW

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Inverse probability weighting (IPW)<latexit sha1_base64="x9MwVMOEAp+s2GhuQc6v5IjE2Eg=">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</latexit>

W

T

Brady Neal

Propensity Scores and IPWY

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Inverse probability weighting (IPW)<latexit sha1_base64="KdwhJ0Jld01DCjPFpk6t3N087eM=">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</latexit>

E[Y (t)] = E<latexit sha1_base64="x9MwVMOEAp+s2GhuQc6v5IjE2Eg=">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</latexit>(T = t)YP (t | W )W

TBrady Neal

Propensity Scores and IPWY

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Inverse probability weighting (IPW)<latexit sha1_base64="KdwhJ0Jld01DCjPFpk6t3N087eM=">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</latexit><latexit sha1_base64="KdwhJ0Jld01DCjPFpk6t3N087eM=">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</latexit>

E[Y (t)] = E<latexit sha1_base64="CtPqyah4M0xn8axhaP4crWih7V8=">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</latexit>(T = t)YP (t | W )W

TBrady Neal

Propensity Scores and IPWY

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Inverse probability weighting (IPW)<latexit sha1_base64="KdwhJ0Jld01DCjPFpk6t3N087eM=">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</latexit>

E[Y (t)] = E<latexit sha1_base64="CtPqyah4M0xn8axhaP4crWih7V8=">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</latexit>(T = t)YP (t | W )W

TBrady Neal

Propensity Scores and IPWY

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Inverse probability weighting (IPW)<latexit sha1_base64="KdwhJ0Jld01DCjPFpk6t3N087eM=">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</latexit>

E[Y (t)] = E<latexit sha1_base64="CtPqyah4M0xn8axhaP4crWih7V8=">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</latexit>(T = t)YP (t | W )See proof in Appendix
A.3 of the course book

Brady Neal

Propensity Scores and IPW

W

T

Y

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Inverse probability weighting (IPW)<latexit sha1_base64="KdwhJ0Jld01DCjPFpk6t3N087eM=">AAATn3icrVhZb9tGEF6lV+IesaO35oWJUiABZFVyUzgvAgLYTouiKRzAV2oaBkVSEiFeJSmrDqHH/sj+hv6Jzny7FEldlINYELmanflmdq7ddS90nThpt/+t3fvs8y++/Or+g62vv/n2u4fbO4/O4mAcmfapGbhBdNEzYtt1fPs0cRLXvggj2/B6rn3eGx3w/PmNHcVO4J8kt6F95RkD3+k7ppEQ6Xr7H/3oUg/GiRl49nM9IdHEs/3kxnBfXGldTT/SdNfuJ5ea3o8MM9Ud3yqwMUdJRsuwpqkelvE03XMsTTcDvx+MfYtsejHV9MgZDJOr6+1Gu9XGn7Y46KhBQ6i/42DnQU3owhKBMMVYeMIWvkho7ApDxPS5FB3RFiHRrkRKtIhGDuZtMRVbJDsmLps4DKKO6DmgX5eK6tNvxowhbZIWl74RSWriB8Vj0bgPqnyzfq3Au0pHCmy28ZbePYXpETURQ6JWyWWcm8v16ONVrDqhNbzCah1aSQgK+8EsrblPb5d+J7RCft4Sp00ji6QiGplEc4kqKawjorf0PPtmiEgY4LNpxFavs3ud1Twf0GdEWAaNY1jKtmrijYqLD802bGUeFzFdjfg3rVDatw6rP1uDg7jLVTDvCVFG4gONysjrdBazazVXopCnS3QxfwIOn7hjogTIfIf85xBHOVdNwmQdBiI5wFpCZFBLIf+G+SybQ4rnLuyJETUN+eZAT6gqK7c7s9Oldw/YEcmnNDeEHvZFk+yxgR0hezMvNsEd0a8Jfpmw0Zyjt1CZHM8mrYmzq0m4Ac04MyzpCenPzCJpXUo0TX12xe/Qb1Pcm/AK53WTZANYKLM4hi5fxb5LfYX94tGTqezZjNKEDg92DJEtI6Ll+iQS6/uRVhGjflyFmtLbhddCoDShnf0zwdjDmjiiY+ie0NiCHpbWqNe1xL6yY1rSyr5x0J0WtWrIP65d2RnntbKcj0rVVFZ2oaktXqoVcwTY7qLXTNUNZU+OsbrpnFU92J93Dc7Rsn0slcenHCsL2TiEVBmXvbh8ncvWsoc6aSKW7PsBcXQh1Vf0GLUZKk1bhfo7UFUSIB4m3p7KCVnvrG0yN5MS3VOecolP1jnvSTzzF2EayCkd3ivOr0M8UvLZbsAdPwV1vdwZKqssG9Eonc2sl3cwsuEfOeKIS7wQvg6xliezr1aY2RT70+D5C4icsYlC2hTdQm4sX/mJaFAvaFBWlf3JkWZKhNNJSONnM85nhMu8I2jxicIVns7mp5UxOCTKVHmsj7msDnMbHHQ+C5y6yjzJISsqmePgql6vlfNyWd5I+npZa5bvZVlJXy/LlBvkpqP6QQysiwq5BHFgCa+QpScVUgH2BEmXNr/fwD4fe+1Y5UWkLDyvkPTQY+SqAlWVbyvtS9AFI+UTlvnzIzx4g/43xQnorn7MZZM7eTOXu/0on+bykzt6Npf07ujfXPJDhaSNvcGZ0VjmXWVNMddxJZfMp6o9wabdJDu/yGpr4tkjL/KebanOpKn9hrNC7o5d9IDmRv15cd/hSu6oOs73yuyMGxe6qAPs/Pyb4qwyUf02O7Xswk6p6Veq2Ld0EuDxgfiJTiEder4p9eZNUfk2MFZVXcTdI+R96qqH6L53xw3VmX23dLIv6zgiDdl3qu5AOberTqqrkO6idzmyrChr4cQ8r0meiYbEJe8JtxWIeY0ux1sf3+XI8j4f4Jya3xY+BXLxJrQZXp4zq++tVbfp1Xe8kN59erroO7HiPUTWueg/tjhV91C5xx8s3Kg5f/eoKjh306eP5vSV8efvaAPcscbEkfPIs3YCaUkrIq667YWQSdSZJsbNKL//tUr3OYk3b1tcyCtP4XVVL7JVT9u63m505v9HtDg422t1fm61371svH6l/n90XzwWT8Vz8tO+eE2VeUx+NcV/tZ3a97XH9Sf1X+p/1I8l672akqmL0l/9/f8Zut11</latexit>

E[Y (t)] = E<latexit sha1_base64="CtPqyah4M0xn8axhaP4crWih7V8=">AAAUcHicrVjpbttGEN64l+sesRP/KNACZWIHcApZld0WCVAICOokKNqkSAAfaS3DoMiVRIhXyZVVm9Bj9UH6s6/RJ+jMt0uR1GkHkWBxOTvzzbEzs7tux76Xqkbj31sr773/wYcfrX689smnn31+e33jznEaDRJHHjmRHyVv2nYqfS+UR8pTvnwTJ9IO2r48afcPeP7kQiapF4WH6jKWZ4HdDb2O59iKSOcbt/5utWXXCzPl9a9iz1GDRI7WLPN5YP0WuTIdv7dCej1Nla3kmbXTcqKwEw1Cl/AfWraydhq1xkMr2y5PbI9+nCeuyFAVyFA9tE7b0o+Gli87ympGHauMcMaIY94FeNFAOVEgx2iJ1+3NgTOsOVjJ4Wdut+Twg1Zsq96kq5J4KvaXTFrAn9s3zV2KxEzelgzdygqdr2816g18rOnBnhlsCfN5FW2s/iNawhWRcMRABEKKUCga+8IWKX1PxZ5oiJhoZyIjWkIjD/NSjMQayQ6ISxKHTdQ+/Xbp7dRQQ3pnzBTSDmnx6S8hSUs8MDwujTug6ifrt0q883RkwGYbL+nZNpgBUZXoEXWZXM55XTn2SZGFj+GLR3bGoLCXTsWjDj19eldkP/9eEqekkUtSCY0covlE1RTWkdBTx5U97yHONvgkjdim+d60yfb5K8HzEX37hGXTOIWlbKslnpuoh9AsYSvz+Fix+Yh/kYfavkVYnbEPHlZVe8G8h0TpiysaVZEX6SznznwuZZBHM3QxvwJHSNwpUSLktUfx84ijmokOYbIOGyvZhS8x8qNukH/BfJ6rMa3nLuxJsWoWssmDntjUTWF3bqdPzzawE5LPaK4HPRyLGtkjgZ0gN/Mo1sCd0NsQbw5sdCboddQdr2eNfOLsqhFuRDPeGEtHQsczt0hblxHNMt9d8QL6Ja17DVHhvK6RbAQLdRan0BWatW9S1+C4BPTLVI5sTqlBRwA7esiWPtEKfRqJ9X1LXqSoH9+gZvT0EbUYKDVo5/gMMQ7gE6/oALqHNHahh6Ut6mR18cjYMapo5dh46D3TWi3kH9eu7nuTWlkuRKVaJiub0NQQ3xuPeQXY7nLUHNPrdMdN4d1owqo27C+6Budo1T6WKtanulYusrEHqSouR3G2n7N82Ued1LCWHPsucTQh1TH0FLUZG01rpfo7MFUSYT0cPAOTE7reWdtwYiYjemAi5ROfrnPecXjmT8K0kVMtRK88vwjxmZHPe32bvhmoi+WOUVlV2YRG2XhmsbyHkUR89IhXXOPFiHUMX+6N/6zSzHWx3w1eOIXIGasM0nXRXeTGbM8PxRb1gi3Kqmo8eaWZkuDsEdN4e8y5TbjM24eWkChc4dl4frR0DZ4SZWQi1sFcXoeFDR46nwvOlsk8zaErSk1wcFUv1sp5OStvNH2xrDvO96qspi+WZcoFctMz/SAF1pslcgrrwBJBKUsPl0hF2BM0Xdv8+zXsC7HXDkxeJMbCkyWSAXqM9ioyVflyqX0KXTAxMWGZP94ighfofyOcgG4ax0JW3SiahdzlW8W0kB/eMLKFZHDD+BaSV0skJfYGb0xjmddLa4q5Xi3l0vm0bE+QtJvk5xddbTX8timKvGe7pjNZZr/hrNC7YxM9oGb2uny3y0+paakPetBenGAznDaGpmPm545daNIx+Jlq7iXt5Tw+EN/ROWKPfp9Xuut1Ufk8PzB1WcbdJ+RH1Befon/eHDc2p+7dytm8quMZacj/RuYWU3D75qw5D+kmemcj65pwp868k5r0qaZHXPqkf7kEsaiy2XiL13c2sr5vRzhpFuf9d4FcvstcD6/Imfk3z2X34fm3tJieHfr10TlSw/sUWeejg0hxZG6Sepc+mLoTc/7uU1Vw7mb370zoq+JP3rK6uCUNiKPg0adlBWlNKyPOu6/FkFHmVJLiblPc4OqVG5nGm7QtLeVVYPCapltJ05XWzte39ib/hzM9ON6v7/1Qb7ze33ry2Px/Z1V8Ke6LHYrTI/GEKvMVxdVZ+Wrlp5VfV17c/W/zi82vN+9p1pVbRuauqHw2v/kfEPEkkg==</latexit>(T = t)YP (t | W )See proof in Appendix
A.3 of the course book<latexit sha1_base64="drjK7+xGMxArDmUZ1XIxVBgk+bo=">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</latexit>⌧ , E[Y (1)Brady Neal

Y (0)] = E

(T = 1)Ye(W )Propensity Scores and IPW

W

T

E

Y


(T = 0)Y
1 e(W )

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Inverse probability weighting (IPW)<latexit sha1_base64="KdwhJ0Jld01DCjPFpk6t3N087eM=">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</latexit>

E[Y (t)] = E<latexit sha1_base64="CtPqyah4M0xn8axhaP4crWih7V8=">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</latexit>(T = t)YP (t | W )W

See proof in Appendix
A.3 of the course book<latexit sha1_base64="drjK7+xGMxArDmUZ1XIxVBgk+bo=">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</latexit>⌧ , E[Y (1)Y (0)] = E

1 X yi⌧ˆ =n1 i:t =1 ê(wi )<latexit sha1_base64="OjJA+nVrmOZoCDNc6NEawen7pdA=">AAAT7HicrVjbbttGEF2lt8S9OfFjX9g4BVJAViU3RYICBgI4CYqiARLAubRRYlAUZRHmrSRl1yH4F30r+toP6lv/pmfOrkRSN8pBLIhczc6cmZ3b7noQ+16adbv/ta588OFHH39y9drWp599/sWX29dvPE+jSeK4z5zIj5KXAzt1fS90n2Ve5rsv48S1g4HvvhicHsr8izM3Sb0oPMouYvd1YJ+E3shz7Ayk4+1/+2M7y/uZPSmsA6s/Smwn7xV5eNwrrH46CY5z70ernwEyC9wwO7P9Yw+MmDXM/WiSOVHgcqbINVycFrf7ThSOokk4hHZOfltYe1UNXcFYrQLTSzX0BGWdluPt3W6nyz9rcdAzg11l/p5E16+1VF8NVaQcNVGBclWoMox9ZasUn1eqp7oqBu21ykFLMPI476pCbUF2Ai4XHDaop3ie4NcrQw3xWzBTSjvQ4uObQNJS3xieIcYjUvVb9FsV3lU6cmKLjRd4DwxmAGqmxqA2yU05N5cb4BM0rDrDGu5xtR5WEpMifnBqax7h7eN3hhXK8wKcLkZDSCUYOaD5oGqK6Ejw1p4X34wZCZt8LkZi9Tq711kt8xE+p8CyMU5pqdhqqUcmLiE1u7RVeHzGdDXiH1ihtm8d1mi2Bo9x16sQ3iNQTtVbjOrI63RWs2s1V2aQiyW6hD8jRwjuFJSIme/Bfx446rnqAFN02IzkCdcSM4M6Bvlnzk+zOUY892hPyqhZzDePemJTWaXdUzt9vAfETiCfY25MPeKLNuxxiZ0we6debJM7wa9z/nJoozNH77AyJZ5trEmyqw3cCDPeDEt7QvtzapG2LgfNMp899Qv1u4h7m16RvG5DNqKFOotT6gpN7A/QV8QvAZ5CFc9OKW3qCGjHmNlyClqpTyOJvu+wipT14xvUHG+fXouJ0qZ28c85xwHXJBGdUPc5xkPqEWkLva6j7ho7ippW8Y3H7rSo1WL+Se3qzjivVeRCVqplsvKAmrrqjlmxREDsrnrNMd1Q9+SUqyvmrBrQ/rJrSI7W7ROpMj71WA2ZjWNK1XHFi8vXuWwt+6yTNmMpvj8BxwGlRoaesjZjo2mrUn+HpkoixsPhOzA5oetdtJ3PzeSgB8ZTPvh0ncueJDO/A9NmTvXpver8OsSHRn66G0jHz0ldL/eclVWXTTDKZzPr5T2OXPpHjyTiGi+mr2Ou5evZ16rMbIr9fvDCBUTJ2MwgbYo+ZG4sX/mR2kUv2EVW1f0pkRZKwtNJjPGtGect4ArvKbWEoEiF57P5ojEGD0ApjMdGnJvWYWmDx843JGffZJ7m0BWVzXFIVa/XKnm5LG80fb3scJbvdVlNXy8rlDPmpmf6QUqslw1yGeMgEkElS48apCLuCZqubf51A/tC7rUTkxeJsfBFg2TAHqNXFZmqfNxoX8YumBifiMxv7+DBM/a/giegy/qxlM0u5c1S7uKdfFrKn1/Ss6VkcEn/lpJvGyRd7g3ejCYyTxtrSrieNHLpfGraE1zsJtPzi662Np8DeFH27KHpTJbZbyQr9O54wB7Q3qg/L+47Usk9U8flXjk946aVLuoRuzz/5jyrnJt+Oz217NFOreknVOxjnARkfKi+xymkh+ejWm/eFFVuAxNT1VXcfSDfRVd9wO57edzYnNn3aif7uo6H0DD9FuYOVHL75qS6Cukyepcj64oaLpyY5zXpM9EYXPqecNGAWNbocrz18V2OrO/zEc+p5W3hfSBXb0Kb4ZU5s/re2nSbXn3Hi/Ee4emz76SG9wGzzmf/cdUzcw/Ve/zhwo1a8ncfVSG5m9+8Maevjj9/RzvhHWsCjpJHn7UzSmtaFXHVbS+mTGbONClvRuX9r1O7z2m8edvSSl4FBu/A9CLX9LSt4+3d3vz/iBYHz/c7vR863ad3du/fM/8/uqq+UjfVbfjprrqPynwCvzqte603rZPWeCfc+XPnr52/NeuVlpHZUbW/nX/+B1hN/IE=</latexit>i

Brady NealT

(T = 1)Ye(W )1 X
n0 i:t =0 1
i

E

Y


(T = 0)Y
1 e(W )yiê(wi )

Propensity Scores and IPW

26 / 35

Questions:
1. What happens if the estimatedpropensity score for some unit is 1 or 0?
2. What happens if the estimatedpropensity score is near 1 or 0?

IPW CATE estimation

Brady Neal

Propensity Scores and IPW

28 / 35

IPW CATE estimation
Not quite as natural with IPW as with COM, so beyond scope of course

Brady Neal

Propensity Scores and IPW

28 / 35

IPW CATE estimation
Not quite as natural with IPW as with COM, so beyond scope of course
Simple extension:<latexit sha1_base64="WNmK867A4qokDcAGsHv/v/sp128=">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</latexit>✓
X1(ti = 1)yi
⌧ˆ(x) =
nx i:x =xê(wi )i

Brady Neal

Propensity Scores and IPW(ti = 0)yi
1 ê(wi )◆28 / 35

IPW CATE estimation
Not quite as natural with IPW as with COM, so beyond scope of course
Simple extension:<latexit sha1_base64="WNmK867A4qokDcAGsHv/v/sp128=">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</latexit>✓
X1(ti = 1)yi
⌧ˆ(x) =
nx i:x =xê(wi )
i(ti = 0)yi
1 ê(wi )◆See, e.g., Abrevaya et al. (2015) and references therein

Brady Neal

Propensity Scores and IPW

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Questions:
1. What is the graphical intuition for how inverse
probability weighting deals with confounding?
2. What do we model in IPW? What did wemodel in COM/GCOM estimation?

Conditional Outcome Modeling
Increasing Data Efficiency
Propensity Scores and IPW
Other Methods

Brady Neal

Other Methods

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Using both conditional outcome models andpropensity score models
Model both µ(t, w) and e(w)<latexit sha1_base64="764gcQDJm+Qf4J616I12HL9cw6U=">AAATIXicrVhLb9tGEF6nr8R9OeqxFzZKgRSgVclN4VwMBLATFEUDOIBfaRwEEkVZhPgqSVl1CP2KXttTf01vRW9F/0xnvh2KpB6kFMSCyNXszDez89pd90LXiZN2+9+tW++9/8GHH92+s/3xJ59+9vnO3cZZHIwjyz61AjeILnrd2HYd3z5NnMS1L8LI7no91z7vjQ55/vzajmIn8E+Sm9B+5XWvfGfgWN2ESC8uvfGDxDQm37zeabZbbfwZi4OODJpK/o6Du3e21KXqq0BZaqw8ZStfJTR2VVfF9HmpOqqtQqK9UinRIho5mLfVVG2T7Ji4bOLoEnVEzyv69VKoPv1mzBjSFmlx6RuRpKG+Fp4+jQeg6jfrNwq8q3SkwGYbb+jdE0yPqIkaErVOLuNcX65HH69m1Qmt4RFW69BKQlDYD1ZpzQN6u/Q7oRXy84Y4bRr1SSqikUU0l6iawjoiemvPs2+GiEQXfDaN2Ooqu6us5vmAPiPC6tI4hqVsq6GeSlx8aLZhK/O4iOlqxF9phdq+KqzBbA0O4q5XwbwnRBmpNzQqI1fpLGbXaq5EkKdLdDF/Ag6fuGOiBMh8h/znEEc5Vy3CZB1dRPIKawmRQS1B/hHzWTaHFM9d2BMjagbyzYGeUCortzuz06V3D9gRyac0N4Qe9oVJ9tjAjpC9mRdNcEf0a4JfFmy05ugtVCbH06Q1cXaZhBvQjDPD0p7Q/sws0talRDPks6t+gn6b4m7CK5zXJskGsFBncQxdvsT+gPoK+8WjJ1PZsxnFhA4PdgyRLSOi5fo0Euv7llYRo35cQU3p7cJrIVBMaGf/TDD2sCaO6Bi6JzTuQw9LG9TrWmpf7JiWtLJvHHSnRa0G8o9rV3fGea0s56NSDcnKA2hqq4eyYo4A2130miXdUPfkGKubzlnVg/151+AcLdvHUnl8yrHqIxuHkCrjsheXr3PZWvZQJyZiyb6/Io4DSA2EHqM2Q9G0Xai/Q6mSAPGw8PYkJ3S9s7bJ3ExKdE885RKfrnPek3jmF8LsIqcu4b3ifBXiE5HPdgPu+Cmo1XJnqKyybESjdDZTLe9gZMM/esQR13ghfB1iLV/NvkZhZl3sd4PnLyByxiaCtC56H7mxfOUnqkm9oElZVfYnR5opEU4nIY3vzzjvEy7zjqDFJwpXeDqbn9bG4IgoU/HYAHNZHeY2OOh8fXBeSuZpDl1RyRwHV3W1Vs7LZXmj6dWy/Vm+l2U1vVqWKdfITUf6QQysixq5BHFgCa+QpSc1UgH2BE3XNr9Ywz4fe+1Y8iISC89rJD30GL2qQKryWa19CbpgJD5hmZ/fwoPX6H9TnIA29WMum2zkzVzu5q18mstPNvRsLult6N9c8k2NpI29wZnRWOZ5bU0x13Etl86nuj3Bpt0kO7/oajPx7JEXec/uS2cyZL/hrNC74wF6gLlWf17cd7iSO1LH+V6ZnXHjQhd1gJ2ff1OcVSbSb7NTyy7s1Jp+oIp9RicBHh+q7+gU0qHn01JvXheVbwNjqeoi7h4h71NXPUL33Rw3lDP7bulkX9bxhDRk36ncgXJuV06qq5A20bscWVdUf+HEPK9Jn4mGxKXvCTc1iHmNLserju9yZH2fD3BOzW8L7wK5eBNaDy/PmdX31rrb9Oo7XkjvAT1d9J1YeI+QdS76j61O5R6q9/jDhRs15+8eVQXnbnqvMaevjD9/R7vCHWtMHDmPPmsnkNa0IuKq214ImUTONDFuRvn9r1W6z2m8edviQl55gncgvciWnrb9eqfZmf8f0eLgbK/V+b7Vfv6w+fiR/P/otvpS3VMPyE/76jFV5jH5le8zv6nf1R+NPxt/Nf5u/KNZb22JzBeq9Nf4738Kp7Eu</latexit>

Brady Neal<latexit sha1_base64="p2tZNeJwIr4C/rDQfRv6yuxP/fU=">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</latexit>

Other Methods

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Using both conditional outcome models andpropensity score models
Model both µ(t, w) and e(w)<latexit sha1_base64="764gcQDJm+Qf4J616I12HL9cw6U=">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</latexit>

Example:<latexit sha1_base64="p2tZNeJwIr4C/rDQfRv6yuxP/fU=">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</latexit>1X⌧ˆ =[µ̂(1, wi )n i<latexit sha1_base64="BLAGQKJnveyRNHVAAQVUIPeR5QM=">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</latexit>

Brady Nealµ̂(0, wi )]Other Methods

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Using both conditional outcome models andpropensity score models
Model both µ(t, w) and e(w)<latexit sha1_base64="764gcQDJm+Qf4J616I12HL9cw6U=">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</latexit>

Example:<latexit sha1_base64="p2tZNeJwIr4C/rDQfRv6yuxP/fU=">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</latexit>1X⌧ˆ =[µ̂(1, ê(wi ))n i<latexit sha1_base64="6lM0RpK0qzA6HaNun2dL5L8dNlA=">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</latexit>

Brady Neal

Other Methodsµ̂(0, 1ê(wi ))]

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Doubly robust methods

Brady Neal

Other Methods

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Doubly robust methods• Model both µ(t, w) and e(w)<latexit sha1_base64="764gcQDJm+Qf4J616I12HL9cw6U=">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</latexit><latexit sha1_base64="p2tZNeJwIr4C/rDQfRv6yuxP/fU=">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</latexit>

Brady Neal

Other Methods

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Doubly robust methods• Model both µ(t, w) and e(w)• Consistent if either µ̂(t, w) or ê(w) is consistent<latexit 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Brady Neal

Other Methods

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Doubly robust methods• Model both µ(t, w) and e(w)• Consistent if either µ̂(t, w) or ê(w) is consistent• Theoretically converge to the estimand at a faster rate than COM/IPW<latexit 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Brady Neal

Other Methods

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Doubly robust methods• Model both µ(t, w) and e(w)• Consistent if either µ̂(t, w) or ê(w) is consistent• Theoretically converge to the estimand at a faster rate than COM/IPW• See Section 7.7 of the course book for references to relevant papers<latexit 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Brady Neal

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MatchingControl

TreatedBrady Neal

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MatchingControl

TreatedBrady Neal

• Space: raw, coarsened,
propensity score

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MatchingControl

TreatedBrady Neal

• Space: raw, coarsened,
propensity score• Different criteria for“close enough”

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Double machine learning
Stage 1:<latexit sha1_base64="TMGQUfwgdCRK6Is0/zKrtPaXtmA=">AAAUb3icrVjpbttGEF67l+tedmKgPwIUbOwANiCrspvCBQIDKWwHRdoUCeA4aS3DoMiVRIhXyVVUh9Bb9UX6s4/RN+jMt0uR1O0gEiwuZ2e+OXZmdtet2PdS1Wj8u7L6wYcfffzJ2qfrn33+xZdfbWzeuUijfuLIl07kR8nrlp1K3wvlS+UpX76OE2kHLV++avVOeP7VG5mkXhSeq5tYXgV2J/TanmMrIl1vrvzdbMmOF2bK672NPUf1Ezlct8zngfVb5Mp09N4M6fUyVbaSV9Zu04nCdtQPXcLfs2xl7TZqjT0r2ylP7AwfzRJXZKgKZKj2rMuW9KOB5cu2so6jtlVGuGLEEe8cvKivnCiQI7TE63RnwBnWHKzk8JnbKTsc26o77qkklor5j5biz82b5C4FYipvU4ZuZYGuN7Yb9QY+1uTgwAy2hfk8jzbX/hFN4YpIOKIvAiFFKBSNfWGLlL6X4kA0REy0K5ERLaGRh3kphmKdZPvEJYnDJmqPfjv0dmmoIb0zZgpph7T49JeQpCUeGB6Xxm1Q9ZP1WyXeWToyYLONN/RsGcyAqEp0ibpILudcVo59UmThj/DFIztjUNhLp+JRm54+vSuyn39viFPSyCWphEYO0XyiagrrSOip48qedxFnG3ySRmzTbG9aZPvsleD5iL49wrJpnMJSttUST0zUQ2iWsJV5fKzYbMS/yENt3zys9sgHD6uqvWDec6L0xFsaVZHn6SznzmwuZZCHU3QxvwJHSNwpUSLktUfx84ijmokOYbIOGyvZgS8x8qNukJ9iPs/VmNZzH/akWDUL2eRBT2zqprA7t9OnZwvYCclnNNeFHo5FjeyRwE6Qm3kUa+BO6G2ANwc2OmP0OuqO17NGPnF21Qg3ohlvhKUjoeOZW6Sty4hmme+++BX6Ja17DVHhvK6RbAQLdRan0BWatT+mrsFxCeiXqRzZnFKDjgB2dJEtPaIV+jQS6/uOvEhRP75BzejpI2oxUGrQzvEZYBzAJ17RPnQPaOxCD0tb1Mnq4sjYMaxo5dh46D2TWi3kH9eu7nvjWlkuRKVaJiuPoakhHhqPeQXY7nLUHNPrdMdN4d1wzKoW7C+6Budo1T6WKtanulYusrELqSouR3G6n9N8OUSd1LCWHPsOcRxDqm3oKWozNprWS/V3Yqokwno4eAYmJ3S9s7bB2ExG9MBEyic+Xee84/DMn4RpI6eaiF55fh7imZHPe32Lvhmo8+UuUFlV2YRG2WhmvryHkUR89IhXXOPFiHUMX74d/VmlmWWx3w9eOIHIGasM0rLoLnJjuufnYpt6wTZlVTWevNJMSXD2iGm8M+LcIVzm7UFLSBSu8Gw0P1y4BqdEGZqItTGX12Fhg4fO54KzaTJPc+iKUmMcXNXztXJeTssbTZ8v647yvSqr6fNlmfIGuemZfpAC6/UCOYV1YImglKXnC6Qi7Amarm3+fQn7Quy1fZMXibHw1QLJAD1GexWZqny20D6FLpiYmLDMH+8QwTfof0OcgG4bx0JW3SqahdzNO8W0kB/cMrKFZHDL+BaSbxdISuwN3ojGMi8W1hRzPV/IpfNp0Z4gaTfJzy+62mr4bVEUec92TWeyzH7DWaF3x2P0gJrZ6/LdLj+lpqU+6EF7cYLNcNoYmI6Znzv2oUnH4GequWe0l/P4RHxP54gD+n1S6a7LovJ5vm/qsox7SMhH1BdP0T9vjxubU/d+5Wxe1XFGGvK/obnFFNy+OWvOQrqN3unIuibciTPvuCZ9qukSlz7p3yxALKpsOt789Z2OrO/bEU6axXn/fSCX7zLL4RU5M/vmueg+PPuWFtOzTb8+OkdqeE+RdT46iBQvzU1S79InE3dizt9DqgrO3ez+nTF9VfzxW1YHt6Q+cRQ8+rSsIK1pZcRZ97UYMsqcSlLcbYobXL1yI9N447alpbwKDN6x6VbSdKX1643tg/H/4UwOLg7rBz/UGy8Otx8/NP/fWRP3xH2xS3E6Eo+pMp9TXJ3Ve6s/rT5d/eXuf1tfb32zZWnW1RUjc1dUPlt7/wPBkyRf</latexit>

W

T

Stage 2:

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Double machine learning
Stage 1:• Fit a model to predict Y from W to get the predicted Ŷ
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T

Stage 2:

Brady Neal

W

Other Methods

Y

34 / 35

Double machine learning
Stage 1:• Fit a model to predict Y from W to get the predicted Ŷ• Fit a model to predict T from W to get the predicted T̂<latexit sha1_base64="lxhbd8LGZH5rDwyoGbq0+OfBZS0=">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</latexit><latexit sha1_base64="TMGQUfwgdCRK6Is0/zKrtPaXtmA=">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</latexit>

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T

Stage 2:

Brady Neal

Other Methods

Y

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Double machine learning
Stage 1:• Fit a model to predict Y from W to get the predicted Ŷ• Fit a model to predict T from W to get the predicted T̂<latexit sha1_base64="lxhbd8LGZH5rDwyoGbq0+OfBZS0=">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</latexit><latexit sha1_base64="TMGQUfwgdCRK6Is0/zKrtPaXtmA=">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</latexit>

W<latexit sha1_base64="gzui9GE6EB9hMYTo7GbcwXD2Pio=">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</latexit>

Stage 2:
Partial out W by fitting a model to predict Y<latexit sha1_base64="uzP6ipARkSragJuxS44DZgofI/k=">AAATI3icrVhLb9tGEF6nr8R9OeqxFzZKgR5oVXJTOBcDAewERdEADuBn4yCQKMoixFdJyqpD6Gf02p76a3oreumh/6Uz3w5FUg9SCmJB5Gp25pvZee2ue6HrxEm7/e/Wnffe/+DDj+7e2/74k08/+3znfuMsDsaRZZ9agRtEF71ubLuOb58mTuLaF2Fkd72ea5/3Roc8f35jR7ET+CfJbWi/8rrXvjNwrG5CpJeXxq5xNewm6eX09U6z3Wrjz1gcdGTQVPJ3HNy/t6WuVF8FylJj5Slb+Sqhsau6KqbPS9VRbRUS7ZVKiRbRyMG8raZqm2THxGUTR5eoI3pe06+XQvXpN2PGkLZIi0vfiCQN9bXw9Gk8AFW/Wb9R4F2lIwU223hL755gekRN1JCodXIZ5/pyPfp4NatOaA2PsVqHVhKCwn6wSmse0Nul3wmtkJ+3xGnTqE9SEY0sorlE1RTWEdFbe559M0QkuuCzacRWV9ldZTXPB/QZEVaXxjEsZVsN9Uzi4kOzDVuZx0VMVyP+SivU9lVhDWZrcBB3vQrmPSHKSL2hURm5Smcxu1ZzJYI8XaKL+RNw+MQdEyVA5jvkP4c4yrlqESbr6CKS11hLiAxqCfKPmM+yOaR47sKeGFEzkG8O9IRSWbndmZ0uvXvAjkg+pbkh9LAvTLLHBnaE7M28aII7ol8T/LJgozVHb6EyOZ4mrYmzyyTcgGacGZb2hPZnZpG2LiWaIZ9d9RP02xR3E17hvDZJNoCFOotj6PIl9gfUV9gvHj2Zyp7NKCZ0eLBjiGwZES3Xp5FY37e0ihj14wpqSm8XXguBYkI7+2eCsYc1cUTH0D2hcR96WNqgXtdS+2LHtKSVfeOgOy1qNZB/XLu6M85rZTkflWpIVh5AU1s9khVzBNjuotcs6Ya6J8dY3XTOqh7sz7sG52jZPpbK41OOVR/ZOIRUGZe9uHydy9ayhzoxEUv2/TVxHEBqIPQYtRmKpu1C/R1KlQSIh4W3Jzmh6521TeZmUqJ74imX+HSd857EM78QZhc5dQXvFeerEJ+KfLYbcMdPQa2WO0NllWUjGqWzmWp5ByMb/tEjjrjGC+HrEGv5avY1CjPrYr8bPH8BkTM2EaR10fvIjeUrP1FN6gVNyqqyPznSTIlwOglp/HDG+ZBwmXcELT5RuMLT2fy0NgZHRJmKxwaYy+owt8FB5+uD80oyT3PoikrmOLiqq7VyXi7LG02vlu3P8r0sq+nVsky5QW460g9iYF3UyCWIA0t4hSw9qZEKsCdourb5cg37fOy1Y8mLSCw8r5H00GP0qgKpyue19iXogpH4hGV+fgsP3qD/TXEC2tSPuWyykTdzudu38mkuP9nQs7mkt6F/c8k3NZI29gZnRmOZF7U1xVzHtVw6n+r2BJt2k+z8oqvNxLNHXuQ9uy+dyZD9hrNC744H6AHmWv15cd/hSu5IHed7ZXbGjQtd1AF2fv5NcVaZSL/NTi27sFNr+oEq9jmdBHh8qL6jU0iHns9KvXldVL4NjKWqi7h7hLxPXfUI3Xdz3FDO7Lulk31Zx1PSkH2ncgfKuV05qa5C2kTvcmRdUf2FE/O8Jn0mGhKXvifc1iDmNbocrzq+y5H1fT7AOTW/LbwL5OJNaD28PGdW31vrbtOr73ghvQf0dNF3YuE9Qta56D+2OpV7qN7jDxdu1Jy/e1QVnLvpg8acvjL+/B3tGnesMXHkPPqsnUBa04qIq257IWQSOdPEuBnl979W6T6n8eZtiwt55QnegfQiW3ra9uudZmf+f0SLg7O9Vuf7VvvFo+aTx/L/o7vqS/VAfUN+2ldPqDKPya+s6Tf1u/qj8Wfjr8bfjX80650tkflClf4a//0PlDmyLA==</latexit>

Brady Neal

Other Methods

TYŶ from T<latexit sha1_base64="yvSwVpXfExC7cKr8YH/Pht10fNQ=">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</latexit>T̂

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Causal trees and forests
Flexible and yield valid confidence intervals (for sampling variability)
Is it okay to post memes
in professional forums?
Yes

Do you have a distribution of educational
ideas committee to consult?

Does your CEO decide what you say?
No

No

Do you have a CEO?

Yes

Yes

Do you have high enough worker credit?

No

No

Can you leave?
Yes

Yes

No
Yes

Brady Neal

Yes

Yes

Other Methods

Yes

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