Chen Avin Ilya Shpitser Judea Pearl Computer Science Department UCLA

1. Chen Avin Ilya Shpitser Judea Pearl Computer Science Department UCLA IDENTIFIABILITY OF PATH-SPECIFIC EFFECTS

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

5. 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”) CAUSAL MODELS AND COUNTERFACTUALS

6. COUNTERFACTUALS: STRUCTURAL SEMANTICS u u W W Z X X=x Z Y Yx(u)=y 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. Functional Bayes Net Probability of Counterfactuals:

7. TOTAL, DIRECT, AND INDIRECT EFFECTS HAVE CONTROLLED-BASED SEMANTICS IN LINEAR MODELS b X 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. 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. 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. GENDER QUALIFICATION HIRING 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)? 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) Y 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. GENDER QUALIFICATION HIRING POLICY IMPLICATIONS (Who cares?) What is the indirect effect of X on Y? The effect of Gender on Hiring if sex discrimination is eliminated. 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. x* z* = Zx*(u) GENERAL PATH-SPECIFIC EFFECTS (Def.) X X W Z W Z Y Y Form a new model, , specific to active subgraph g Definition: g-specific effect

17. EFFECT-INVARIANT 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 W Z Z’ Z” Y

20. AZT EXAMPLE REVISITED AZT AZT Headaches Pneumonia Headaches Pneumonia Painkillers Painkillers Antibiotics Antibiotics 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 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.