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.
(Retrospective Intervention Effects; RIE)
• (outcome)
•
• (PSM)
•
(MSE)
2

3.
•
• ( )
˙ ˙ ˙
3

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.
Y
Aj j
• 2×2×2
{A1,A2,A3}
W
U
• (A)
×
W //
**
%%
Y
A1
44
A2
99
...
??
U //
55
::
??
AJ
CC
5

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.
: (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.
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.
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.
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.
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.
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.
RIE (
)
Aj ⊥ (Yi(a,A−ji),Yi(a′
,A−ji))′
|(A−ji,W)
12

14.
OLS Matching
1. (homogenous)
2.
3. DGP
•
Direct matching (Ho et al. 2007 )
• →
13

15.
RIE

16.
RIE
j (Aj) RIE
ψj = E[Y(aj,A−j)]
counterfactual
− E[Y]
observed
• Aj aj ( Aj )
14

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.
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.
ˆgj(aj|Wi,A−ji)
•
• logistic KRLS BART
• v-fold Cross-validation
1.
2. (MSE)
⇒ Super Learner algorithm
17

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.
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.
Ensemble
w ⇒ Ensemble IPW
ˆgj(aj|Wi,A−ji) =
C
∑
c=1
wc∗
j ˆgc
j (aj|Wi,A−ji).
20

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.
3 2 1 0 1 2
0.00.10.20.30.4
W1
Propensityscore
Propensity score over W1
21

25.
3 2 1 0 1 2
050100150
W1
Y(1)(filled)andY(0)(hollow)
Potential outcomes over W1
21

26.
3 2 1 0 1 2
050100150
W1
Treated(filled)andcontrol(hollow)outcomes
Observed data
(after treatment assignment)
21

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.
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.
Number of noise covariates
S.E.
0 5 10
0.91.01.1
OLS
Matching
Naive IPW
Ensemble IPW
•
23

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.
1 ( )
• 0 ∼ 3
• 114
24

32.
1 ( )
• {0,1}
• Employment:
• Security:
• Confidence:
• Depression:
• Excom.peers:
• Ties to commander:
24

33.
1. WLS:
2. Naïve IPW: Logistic
3. Matching:
4. Ensemble IPW ←
25

34.
•
• BART ...
26

35.
( )
/11C2:5385,12945,556C51D19125183129453558491
27

36.
( )
/11C2:5385,12945,556C51D19125183129453558491
28

37.
Ensemble
29

38.
RIE
C2:5385,12945,556C51D19125183129453558491
30

39.
•
•
• HPC 20
•
31