1. Chen Avin Ilya Shpitser Judea Pearl Computer Science Department UCLA
IDENTIFIABILITY OF PATH-SPECIFIC EFFECTS
Chen Avin
Ilya Shpitser
Judea Pearl
Computer Science Department
UCLA
The subject of my lecture this evening is CAUSALITY. It is not an easy topic to speak about, but it is a fun topic to speak about. It is not easy because, like religion, sex and intelligence, causality was meant to be practiced, not analyzed. And it is fun, because, like religion, sex and intelligence, emotions run high, examples are plenty, there are plenty of interesting people to talk to, and above all, an exhilarating experience of watching our private thoughts magnified under the microscope of formal analysis.

2. QUESTIONS ASKED Why path-specific effects?
What are the semantics of path-specific effects
(in nonlinear and nonparametric models)?
What are the policy implications of path-specific effects?
When can path-specific effects be estimated consistently from experimental or nonexperimental data?
Can these conditions be verified from accessible causal knowledge, i.e., graphs?

3. WHY DECOMPOSE
EFFECTS?
Direct (or indirect) effect may be more transportable.
Indirect effects may be prevented or controlled.
Direct (or indirect) effect may be forbidden
Pill 
Pregnancy + +
Thrombosis
Gender
Qualification
Hiring

4. EFFECT-DECOMPOSITION
IN LINEAR MODELS b X Z a c Y a bc
Definition:

5. CAUSAL MODELS AND COUNTERFACTUALS
Definition: A causal model is a 3-tuple
M = V,U,F
(i) V = {V1…,Vn} endogenous variables,
(ii) U = {U1,…,Um} background variables (unit)
F = set of n functions,
The sentence: “Y would be y (in unit u), had X been x,”
denoted Yx(u) = y, is the solution for Y in a mutilated model
Mx, with the equations for X replaced by X = x.
(“unit-based potential outcome”)

6. COUNTERFACTUALS: STRUCTURAL SEMANTICS u Y Z W X u Yx(u)=y Z W X=x
Notation: Yx(u) = y
Y has the value y in the solution to a mutilated system of equations, where the equation for X is replaced by a constant X=x. u Y Z W X u
Yx(u)=y Z W
X=x
Functional
Bayes Net
Probability of Counterfactuals:

7. TOTAL, DIRECT, AND INDIRECT EFFECTS HAVE CONTROLLED-BASED SEMANTICS IN LINEAR MODELSX Z
z = bx + 1
y = ax + cz + 2 a c Y
a + bc a bc

8. CONTROLLED-BASED SEMANTICS NONTRIVIAL IN NONLINEAR MODELS
(even when the model is completely specified) X Z
z = f (x, 1)
y = g (x, z, 2) Y
Dependent on z?
Void of operational meaning?

9. (FORMALIZING DISCRIMINATION)
LEGAL DEFINITIONS
OF DIRECT EFFECT
(FORMALIZING DISCRIMINATION)
``The central question in any employment-discrimination case is whether the employer would have taken the same action had the employee been of different race (age, sex, religion, national origin etc.) and everything else had been the same’’
[Carson versus Bethlehem Steel Corp. (70 FEP Cases 921, 7th Cir. (1996))]
x = male, x = female
y = hire, y = not hire
z = applicant’s qualifications
NO DIRECT EFFECT

10. AVERAGE DIRECT EFFECTS
NATURAL SEMANTICS OF
AVERAGE DIRECT EFFECTS
Robins and Greenland (1992) – “Pure” X Z
z = f (x, u)
y = g (x, z, u) Y
Average Direct Effect of X on Y:
The expected change in Y, when we change X from x0 to x1 and, for each u, we keep Z constant at whatever value it attained before the change.
In linear models, DE = Controlled Direct Effect

11. POLICY IMPLICATIONS (Who cares?) What is the direct effect of X on Y?
Is employer guilty of sex-discrimination given data
on (X,Y,Z)?
GENDER
QUALIFICATION
HIRING X Z
CAN WE IGNORE
THIS LINK? f Y

12. NATURAL SEMANTICS OF INDIRECT EFFECTS X Z z = f (x, u) y = g (x, z, u)
Indirect Effect of X on Y:
The expected change in Y when we keep X constant, say at x0, and let Z change to whatever value it would have attained had X changed to x1.
In linear models, IE = TE - DE

13. POLICY IMPLICATIONS (Who cares?)
What is the indirect effect of X on Y?
The effect of Gender on Hiring if sex discrimination
is eliminated.
GENDER
QUALIFICATION
HIRING X Z
IGNORE f Y

14. SEMANTICS AND IDENTIFICATION OF NESTED COUNTERFACTUALS
Consider the quantity
Given M, P(u), Q is well defined
Given u, Zx*(u) is the solution for Z in Mx*, call it z
is the solution for Y in Mxz
Can Q be estimated from data?
Experimental: nest-free expression
Nonexperimental: subscript-free expression

15. IDENTIFICATION IN MARKOVIAN MODELS Corollary 3:
The average direct effect in Markovian models is
identifiable from nonexperimental data, and it is given by
where S stands for any sufficient set of covariates.
Example:
S =  X Z Y

16. GENERAL PATH-SPECIFIC
EFFECTS (Def.) X x* X W Z W Z
z* = Zx* (u) Y Y
Form a new model, , specific to active subgraph g
Definition: g-specific effect

17. EFFECT-INVARIANT Rule 1 Rule 2
Here we discuss the two graph transformations which we use in our proof. These rules do not change the path-specific effect. Dashed arrows represent directed _paths_.
Rule 1
Rule 2

18. MAIN RESULT Applying the two rules results in one of two cases:
Case 1: we obtain a ‘kite pattern.’ Then the path-specific effect is not identifiable. Z
R - Recanting witness Y

19. MAIN RESULT (Cont.)
Case 2: all blocked edges emanate from the root node. Then the effect is identifiable. X
Here we can say that identifiable path-specific effects are related to natural direct and indirect effects. All of these effects only
‘cut’ root-emanating edges. W Z Z’ Z” Y

20. AZT EXAMPLE REVISITED
AZT
AZT
Headaches
Pneumonia
Headaches
Pneumonia
Painkillers
Antibiotics
Painkillers
Antibiotics
This slide traces the application of the two rules for the two path-specific effects in the original example graph, showing how we obtain a root-emanating edge in the first case, and a kite pattern in the second case.
Survival
Survival
Painkiller contribution to the total effect of AZT on survival
Antibiotics contribution to the total effect of AZT on survival

21. RECANTING WITNESS
AZT
R-Recanting Witness
Headaches
Pneumonia
Painkillers
Antibiotics
R behaves as I
Give the intuition where is the problem with the kite structure.
Survival
Antibiotics contribution to the total effect of AZT on survival
R behaves as II
P(RX,RX*) is not experimentally identifiable

22. SUMMARY OF RESULTS
Formal semantics of path-specific effects, based on signal blocking, instead of value fixing.
Path-analytic techniques extended to nonlinear and nonparametric models.
Meaningful (graphical) conditions for estimating effects from experimental and nonexperimental data.
Graphical techniques of inferring effects of policies involving signal blocking.