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Samii et al. (2016)
Retrospective Causal Inference with Machine
Learning Ensembles: An Application to
Anti-recidivism Poli...
(Retrospective Intervention Effects; RIE)
• (outcome)
•
• (PSM)
•
(MSE)
2
•
• ( )
˙ ˙ ˙
3
˙ ?
• ATE (+ATT/ATC)
• ATE E[Y(Treated)]−E[Y(Controlled)]
• RIE E[Y(a)]−E[Y]
• RIE (a)
˙ ˙ ˙ ˙
⇒ ATE
• E[Y(a)] Matching
4
Y
Aj j
• 2×2×2
{A1,A2,A3}
W
U
• (A)
×
W //
**
%%

Y
A1
44
A2
99
...
??
U //
55
::
??
AJ
CC
5
Aji i j
• Aji = a a′
• τji(a,a′) =
Yi(a,A−j)−Yi(a′,A−j)
• j
( )
Q Aj
W //
**
%%

Y
A1
44
A2
99
...
??
U //
55
::
??
AJ
CC
5
: (W,A−j) OK
set.seed(19861008)
W - rnorm(1000, 0, 3)
W - rnorm(1000, 0, 3)
A1 - 1 + 2 * W + 3 * U + rnorm(1000, 0, 0.1)
A...
A1 ( = 2) ...
 summary(lm(Y ~ A1))
Estimate Std. Error t value Pr(|t|)
(Intercept) 11.56375 0.37966 30.46 2e-16
A1 13.3887...
A2,3 ...
 summary(lm(Y ~ A1 + A2 + A3))
Estimate Std. Error t value Pr(|t|)
(Intercept) 3.02025 0.01212 249.219  2e-16
A1 ...
W
 summary(lm(Y ~ A1 + W))
Estimate Std. Error t value Pr(|t|)
(Intercept) 12.359611 0.037851 326.5 2e-16
A1 12.664882 0.0...
A2,3,W
 summary(lm(Y ~ A1 + A2 + A3 + W))
Estimate Std. Error t value Pr(|t|)
(Intercept) 5.00778 0.02246 223.00 2e-16
A1 ...
U
 summary(lm(Y ~ A1 + A2 + A3 + W + U))
Estimate Std. Error t value Pr(|t|)
(Intercept) 4.96969 0.12074 41.159  2e-16
A1 ...
RIE (
)
Aj ⊥ (Yi(a,A−ji),Yi(a′
,A−ji))′
|(A−ji,W)
12
OLS Matching
1. (homogenous)
2.
3. DGP
•
Direct matching (Ho et al. 2007 )
• →
13
RIE
RIE
j (Aj) RIE
ψj = E[Y(aj,A−j)]
counterfactual
− E[Y]
observed
• Aj aj ( Aj )
14
1. A = a Y = Y(a)
• SUTVA
•
2. aj Y(aj,A−ji) ⊥ Aj|(W,A−j)
•
3. aj Pr[Aj = aj|W,A−j]  b (b )
• (overlap)
• overlap
15
RIE IPW
ψIPW
=
1
N
N
∑
i=1
I(aj)
ˆgj(aj|Wi,A−ji)
Yi −Y.
• I(aj): Aj = aj 1 Aj ̸= aj 0
• ˆgj(aj|Wi,A−ji): Pr[Aj = aj|Wi,A−j...
ˆgj(aj|Wi,A−ji)
•
• logistic KRLS BART
• v-fold Cross-validation
1.
2. (MSE)
⇒ Super Learner algorithm
17
Cross-validation
: (MSE)
ℓc
j =
1
N
N
∑
i=1
I(Aji = aj)− ˆg
c,v(i)
j (aj|Wi,A−ji)
2
• ˆgc,v(i): c index v v-fold CV
sub-sa...
Ensemble
(w)
(w1∗
j ,...,wC∗
j ) = arg min(w1∗
j ,...,wC∗
j )
1
N
N
∑
i=1
I(Aji = aj)−
C
∑
c=1
wc
j ˆg
c,v(i)
j (aj|Wi,A−j...
Ensemble
w ⇒ Ensemble IPW
ˆgj(aj|Wi,A−ji) =
C
∑
c=1
wc∗
j ˆgc
j (aj|Wi,A−ji).
20
Y(0) = W1 +0.5(W1 −min(W1))2
+ε0,
Y(1) = W1 +0.75(W1 −min(W1))2
+0.75(W1 −min(W1))3
+ε1,
Pr[A = 1|W1] = logit−1
(−0.5+0.75...
3 2 1 0 1 2
0.00.10.20.30.4
W1
Propensityscore
Propensity score over W1
21
3 2 1 0 1 2
050100150
W1
Y(1)(filled)andY(0)(hollow)
Potential outcomes over W1
21
3 2 1 0 1 2
050100150
W1
Treated(filled)andcontrol(hollow)outcomes
Observed data
(after treatment assignment)
21
(Nc = 5)
1. Logistic regression
2. t-regularized logistic regression
3. Kernal regularized least squares (KRLS)
4. Bayesia...
Number of noise covariates
Bias
0 5 10
0.050.050.100.150.20
OLS
Matching
Naive IPW
Ensemble IPW
• Matching Ensemble IPW
• ...
Number of noise covariates
S.E.
0 5 10
0.91.01.1
OLS
Matching
Naive IPW
Ensemble IPW
•
23
Number of noise covariates
RMSE
0 5 10
0.000.050.100.150.200.25
OLS
Matching
Naive IPW
Ensemble IPW
• Ensemble IPW
• Noise...
1 ( )
• 0 ∼ 3
• 114
24
1 ( )
• {0,1}
• Employment:
• Security:
• Confidence:
• Depression:
• Excom.peers:
• Ties to commander:
24
1. WLS:
2. Naïve IPW: Logistic
3. Matching:
4. Ensemble IPW ←
25
•
• BART ...
26
( )
/11C2:5385,12945,556C51D19125183129453558491
27
( )
/11C2:5385,12945,556C51D19125183129453558491
28
Ensemble
29
RIE
C2:5385,12945,556C51D19125183129453558491
30
•
•
• HPC 20
•
31
Review: Cyrus, Samii, Laura Paler, and Sarah Zukerman Daly. 2016. “Retrospective Causal Inference with Machine Learning En...
Review: Cyrus, Samii, Laura Paler, and Sarah Zukerman Daly. 2016. “Retrospective Causal Inference with Machine Learning En...
Review: Cyrus, Samii, Laura Paler, and Sarah Zukerman Daly. 2016. “Retrospective Causal Inference with Machine Learning En...
Review: Cyrus, Samii, Laura Paler, and Sarah Zukerman Daly. 2016. “Retrospective Causal Inference with Machine Learning En...
Review: Cyrus, Samii, Laura Paler, and Sarah Zukerman Daly. 2016. “Retrospective Causal Inference with Machine Learning En...
Review: Cyrus, Samii, Laura Paler, and Sarah Zukerman Daly. 2016. “Retrospective Causal Inference with Machine Learning En...
Review: Cyrus, Samii, Laura Paler, and Sarah Zukerman Daly. 2016. “Retrospective Causal Inference with Machine Learning En...
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Review: Cyrus, Samii, Laura Paler, and Sarah Zukerman Daly. 2016. “Retrospective Causal Inference with Machine Learning Ensembles: An Application to Anti-recidivism Policies in Colombia.” Political Analysis. 22 (4) pp. 434-456

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社会科学方法論研究会 (2018年3月9日)報告資料

Title: Review: Cyrus, Samii, Laura Paler, and Sarah Zukerman Daly. 2016. “Retrospective Causal Inference with Machine Learning Ensembles: An Application to Anti-recidivism Policies in Colombia.” Political Analysis. 22 (4) pp. 434-456
Data: Mar. 9, 2018
Location: Social Science Methodology Workshop, Osaka University

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Review: Cyrus, Samii, Laura Paler, and Sarah Zukerman Daly. 2016. “Retrospective Causal Inference with Machine Learning Ensembles: An Application to Anti-recidivism Policies in Colombia.” Political Analysis. 22 (4) pp. 434-456

  1. 1. Samii et al. (2016) Retrospective Causal Inference with Machine Learning Ensembles: An Application to Anti-recidivism Policies in Colombia 2018/03/09 0
  2. 2. (Retrospective Intervention Effects; RIE) • (outcome) • • (PSM) • (MSE) 2
  3. 3. • • ( ) ˙ ˙ ˙ 3
  4. 4. ˙ ? • ATE (+ATT/ATC) • ATE E[Y(Treated)]−E[Y(Controlled)] • RIE E[Y(a)]−E[Y] • RIE (a) ˙ ˙ ˙ ˙ ⇒ ATE • E[Y(a)] Matching 4
  5. 5. Y Aj j • 2×2×2 {A1,A2,A3} W U • (A) × W // ** %% Y A1 44 A2 99 ... ?? U // 55 :: ?? AJ CC 5
  6. 6. Aji i j • Aji = a a′ • τji(a,a′) = Yi(a,A−j)−Yi(a′,A−j) • j ( ) Q Aj W // ** %% Y A1 44 A2 99 ... ?? U // 55 :: ?? AJ CC 5
  7. 7. : (W,A−j) OK set.seed(19861008) W - rnorm(1000, 0, 3) W - rnorm(1000, 0, 3) A1 - 1 + 2 * W + 3 * U + rnorm(1000, 0, 0.1) A2 - 2 + 3 * W + 4 * U + rnorm(1000, 0, 0.1) A3 - 3 + 4 * W + 5 * U + rnorm(1000, 0, 0.1) Y - 5 + 1 * W + 2 * A1 + 3 * A2 + 4 * A3 + rnorm(1000, 0, 0.1) 6
  8. 8. A1 ( = 2) ... summary(lm(Y ~ A1)) Estimate Std. Error t value Pr(|t|) (Intercept) 11.56375 0.37966 30.46 2e-16 A1 13.38875 0.03474 385.39 2e-16 ...??? 7
  9. 9. A2,3 ... summary(lm(Y ~ A1 + A2 + A3)) Estimate Std. Error t value Pr(|t|) (Intercept) 3.02025 0.01212 249.219 2e-16 A1 -0.31520 0.04251 -7.414 2.62e-13 A2 2.71173 0.08256 32.844 2e-16 A3 5.62086 0.04165 134.946 2e-16 ......??? 8
  10. 10. W summary(lm(Y ~ A1 + W)) Estimate Std. Error t value Pr(|t|) (Intercept) 12.359611 0.037851 326.5 2e-16 A1 12.664882 0.004146 3054.7 2e-16 W 4.682613 0.014818 316.0 2e-16 .........??? 9
  11. 11. A2,3,W summary(lm(Y ~ A1 + A2 + A3 + W)) Estimate Std. Error t value Pr(|t|) (Intercept) 5.00778 0.02246 223.00 2e-16 A1 2.00331 0.02937 68.22 2e-16 A2 3.02114 0.02755 109.66 2e-16 A3 3.98110 0.02286 174.17 2e-16 W 1.00791 0.01120 89.95 2e-16 !! 10
  12. 12. U summary(lm(Y ~ A1 + A2 + A3 + W + U)) Estimate Std. Error t value Pr(|t|) (Intercept) 4.96969 0.12074 41.159 2e-16 A1 2.00807 0.03291 61.021 2e-16 A2 3.02668 0.03252 93.058 2e-16 A3 3.98850 0.03247 122.848 2e-16 W 0.95214 0.17403 5.471 5.66e-08 U -0.07346 0.22874 -0.321 0.748 ... ( ) 11
  13. 13. RIE ( ) Aj ⊥ (Yi(a,A−ji),Yi(a′ ,A−ji))′ |(A−ji,W) 12
  14. 14. OLS Matching 1. (homogenous) 2. 3. DGP • Direct matching (Ho et al. 2007 ) • → 13
  15. 15. RIE
  16. 16. RIE j (Aj) RIE ψj = E[Y(aj,A−j)] counterfactual − E[Y] observed • Aj aj ( Aj ) 14
  17. 17. 1. A = a Y = Y(a) • SUTVA • 2. aj Y(aj,A−ji) ⊥ Aj|(W,A−j) • 3. aj Pr[Aj = aj|W,A−j] b (b ) • (overlap) • overlap 15
  18. 18. RIE IPW ψIPW = 1 N N ∑ i=1 I(aj) ˆgj(aj|Wi,A−ji) Yi −Y. • I(aj): Aj = aj 1 Aj ̸= aj 0 • ˆgj(aj|Wi,A−ji): Pr[Aj = aj|Wi,A−ji] ⇐ ⇒ Aj = aj 16
  19. 19. ˆgj(aj|Wi,A−ji) • • logistic KRLS BART • v-fold Cross-validation 1. 2. (MSE) ⇒ Super Learner algorithm 17
  20. 20. Cross-validation : (MSE) ℓc j = 1 N N ∑ i=1 I(Aji = aj)− ˆg c,v(i) j (aj|Wi,A−ji) 2 • ˆgc,v(i): c index v v-fold CV sub-sample (hold-out) index (v(i) i v) ⇒ Aj 19
  21. 21. Ensemble (w) (w1∗ j ,...,wC∗ j ) = arg min(w1∗ j ,...,wC∗ j ) 1 N N ∑ i=1 I(Aji = aj)− C ∑ c=1 wc j ˆg c,v(i) j (aj|Wi,A−ji) 2 , C ∑ c=1 wc j = 1, wc j 0. • MSE (ℓc j ) w 20
  22. 22. Ensemble w ⇒ Ensemble IPW ˆgj(aj|Wi,A−ji) = C ∑ c=1 wc∗ j ˆgc j (aj|Wi,A−ji). 20
  23. 23. Y(0) = W1 +0.5(W1 −min(W1))2 +ε0, Y(1) = W1 +0.75(W1 −min(W1))2 +0.75(W1 −min(W1))3 +ε1, Pr[A = 1|W1] = logit−1 (−0.5+0.75W1 −0.5[W1 −mean(W1)]2 ), W,ε ∼ Normal(0,1). (1) DGP • • • A (0 → 1) Y(1) Y(0)+X 21
  24. 24. 3 2 1 0 1 2 0.00.10.20.30.4 W1 Propensityscore Propensity score over W1 21
  25. 25. 3 2 1 0 1 2 050100150 W1 Y(1)(filled)andY(0)(hollow) Potential outcomes over W1 21
  26. 26. 3 2 1 0 1 2 050100150 W1 Treated(filled)andcontrol(hollow)outcomes Observed data (after treatment assignment) 21
  27. 27. (Nc = 5) 1. Logistic regression 2. t-regularized logistic regression 3. Kernal regularized least squares (KRLS) 4. Bayesian additive regression trees (BART) 5. v-support vector machine (SVM) 1. OLS 2. Naïve IPW: logistic regression 3. Matching: Mahalanobis Normal(0,1) noise 0 ∼ 10 22
  28. 28. Number of noise covariates Bias 0 5 10 0.050.050.100.150.20 OLS Matching Naive IPW Ensemble IPW • Matching Ensemble IPW • Ensemble IPW Noise ( ) 23
  29. 29. Number of noise covariates S.E. 0 5 10 0.91.01.1 OLS Matching Naive IPW Ensemble IPW • 23
  30. 30. Number of noise covariates RMSE 0 5 10 0.000.050.100.150.200.25 OLS Matching Naive IPW Ensemble IPW • Ensemble IPW • Noise RMSE × 23
  31. 31. 1 ( ) • 0 ∼ 3 • 114 24
  32. 32. 1 ( ) • {0,1} • Employment: • Security: • Confidence: • Depression: • Excom.peers: • Ties to commander: 24
  33. 33. 1. WLS: 2. Naïve IPW: Logistic 3. Matching: 4. Ensemble IPW ← 25
  34. 34. • • BART ... 26
  35. 35. ( ) /11C2:5385,12945,556C51D19125183129453558491 27
  36. 36. ( ) /11C2:5385,12945,556C51D19125183129453558491 28
  37. 37. Ensemble 29
  38. 38. RIE C2:5385,12945,556C51D19125183129453558491 30
  39. 39. • • • HPC 20 • 31

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