CSDS 600-101 Special Topics: Machine Learning and
Causal Inference (2023 Fall)
Homework 3
Instructor: Jing Ma (jing.ma5@case.edu)
TA: Cerag Oguztuzun (cerag.oguztuzun@case.edu) Due:
Sat 11/18/2023 (23:59 PM EST Time)
Note: The assignment must be submitted electronically on Canvas. Please submit your solution
as a pdf file with name “HW3_Lastname_Firstname”. *The submissions will not be accepted
after the deadline*.
Problem 1 (25 Pts)
Given the causal graph as follows. X represents the amount of time a student spends in an
after-school remedial program, H the amount of homework a student does, and Y a student’s
score on the exam.
Figure 1: Causal model for Problem 1. We assume that all U factors
are independent. Let us consider a student named Joe, for whom we measure X = 0.5, H = 1,
and Y = 1.5.
• (5 pts) Compute the values of the exogenous U variables for Joe.
• (5 pts) Suppose we want to know this counterfactual: "What would Joe’s score have been
had he doubled his homework". Then in the Abduction-Action-Prediction process, what
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would be the modified causal model? Please draw this causal graph and write the
modified structural equation.
• (5 pts) Compute the above counterfactual of Joe’s score had he doubled his homework.
• (5 pts) Compute the counterfactual of Joe’s score had he doubled his amount of time in
after-school remedial program.
• (5 pts) Suppose X is a sensitive feature, in order to achieve counterfactual fairness, what
variables should you include to predict Y ? List them and explain.
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Answer:
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Problem 2 (20 Pts)
Please read the following material:
(Counterfactuals in Linear Models) In nonparametric models, counterfactual quantities of the
form E[YX←x|Z = e] may not be identifiable, even if we have the luxury of running
experiments. In fully linear models, however, things are much easier.
Theory: Let τ be the slope of the total effect of X on Y ,
τ = E[Y |do(x + 1)] − E[Y |do(x)],
then, for any evidence Z = e, we have
E[YX←x|Z = e] = E[Y |Z = e] + τ(x − E[X|Z = e]).
This provides an intuitive interpretation of counterfactuals in linear models: E[YX←x|Z = e] can
be computed by first calculating the best estimate of Y conditioned on the evidence e, E[Y
|e], and then adding to it whatever change is expected in Y when X is shifted from its current
best estimate, E[X|Z = e], to its hypothetical value, x.
• (5 pts) Describe how the parameters a, b, c in Figure 1 can be estimated from
observational data.
• (5 pts) For the causal model in Figure 1, calculate τ. • (5 pts) For the causal model in Figure
1, compute the "effect of treatment on the treated group (ETT)": E[YX←1 − YX←0|X = 1].
How is it compared with τ? • (5 pts) In the following linear model in Figure 2, find the
causal effect of college on those students whose salary is Y = 1. [Hint: that is E[YX←1 −
YX←0|Y = 1].]
Figure 2: Causal model 2.
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Answer:
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Problem 3 (25 Pts)
Please read the following material about direct and indirect causal effects:
A typical mediation problem takes the form:
T = fT (uT ), M = fM(T,uM), Y = fY (T,M,uY ),
where T is treatment, M is mediator, and Y is outcome, fT , fM, and fY are arbitrary structural
functions, and UT , UM, UY represent exogenous variables. Four types of effects can be defined
for the transition from T = 0 to T = 1.
(A) Total effect.
TE = E[Y1 − Y0] = E[Y |do(T = 1)] − E[Y |do(T = 0)]
TE measures the expected increase in Y as the treatment changes from T = 0 to T = 1, while
the mediator is allowed to track the change in T naturally, as dictated by the function fM.
(Here, the potential outcome Yt is actually short for YT←t) (B) Controlled direct effect.
CDE(m) = E[Y1,m − Y0,m] = E[Y |do(T = 1,M = m)] − E[Y |do(T = 0,M = m)].
CDE measures the expected increase in Y as the treatment changes from T = 0 to T = 1, while
the mediator is set to a specified level M = m uniformly over the entire population.
(C) Natural direct effect.
NDE = E[Y1,M0 − Y0,M0]. NDE measures the expected increase
in Y as the treatment changes from T = 0 to T = 1, while the mediator is set to whatever value
it would have attained (for each individual) prior to the change, that is, under T = 0.
(D) Natural indirect effect.
NIE = E[Y0,M1 − Y0,M0].
NIE measures the expected increase in Y when the treatment is held constant, at T = 0, and M
changes to whatever value it would have attained (for each individual) under T = 1. It
captures, therefore, the portion of the effect that can be explained by mediation alone, while
disabling the capacity of Y to respond to X.
In linear systems, we have TE = NDE + NIE.
• (10 pts) Consider the linear structural model:
use the above definition, compute TE, NDE, and NIE.
• (5 pts) Repeat problem 3.1 assuming that uy is correlated with um.
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• (10 pts) Consider the non-linear structural model:
use the above definition, compute TE, NDE, and NIE. Do we still have TE = NDE + NIE?
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Answer:
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Problem 4 (30 Pts)
This is a coding task. See the attached .ipynb file.
Answer:
.ipynb file is submitted via Canvas.
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