THE OVERRIDING THEME

1. THE OVERRIDING THEME • Define Q(M) as a counterfactual expression • Determine conditions for the reduction • If reduction is feasible, Q is inferable. • Demonstrated on three types of queries: Q1: P(y|do(x)) Causal Effect (= P(Yx=y)) Q2: P(Yx = y | x, y) Probability of necessity Q3: Direct Effect

2. OUTLINE • Modeling: Statistical vs. Causal • Causal Models and Identifiability • Inference to three types of claims: • Effects of potential interventions • Claims about attribution (responsibility) • Claims about direct and indirect effects • Actual Causation and Explanation • Robustness of Causal Claims

3. ROBUSTNESS: MOTIVATION a In linear systems: y = ax + u cov (x,u) = 0 a is identifiable. a = Ryx Genetic Factors (unobserved) u x y Smoking Cancer

4. ROBUSTNESS: MOTIVATION Genetic Factors (unobserved) u x y Smoking Cancer The claim a = Ryx is sensitive to the assumption cov (x,u) = 0. a is non-identifiable if cov (x,u) ≠ 0.

5. ROBUSTNESS: MOTIVATION Z Price of Cigarettes b x a is identifiable, even if cov (x,u) ≠ 0 Genetic Factors (unobserved) u a y Smoking Cancer Z – Instrumental variable; cov(z,u) = 0

6. ROBUSTNESS: MOTIVATION Z Price of Cigarettes b x Genetic Factors (unobserved) u a y Smoking Cancer Claim “a = Ryx”is likely to be true

7. ROBUSTNESS: MOTIVATION g Z2 Peer Pressure Invoking several instruments If a0= a1 = a2, claim “a = a0” is more likely correct Z1 Genetic Factors (unobserved) u Price of Cigarettes b a y x Smoking Cancer

8. ROBUSTNESS: MOTIVATION Z3 Anti-smoking Legislation Zn Z1 Genetic Factors (unobserved) u Price of Cigarettes b a g Z2 y x Peer Pressure Smoking Cancer Greater surprise:a1 = a2 = a3….= an = q Claim a = q is highly likely to be correct

9. ROBUSTNESS: MOTIVATION Given a parameter a in a general graph a y x Assume we have several independent estimands of a, and a1 = a2 = …an Find the degree to which a is robust to violations of model assumptions

10. ROBUSTNESS: ATTEMPTED FORMULATION Bad attempt: Parameter a is robust (over identifies) f1, f2: Two distinct functions if:

11. ROBUSTNESS: MOTIVATION b s Symptom Genetic Factors (unobserved) u a x y Smoking Cancer Is a robust if a0 = a1?

12. ROBUSTNESS: MOTIVATION s Symptom Genetic Factors (unobserved) u b a x y Smoking Cancer • Symptoms do not act as instruments • remains non-identifiable if cov (x,u) ≠ 0 Why? Taking a noisy measurement (s) of an observed variable (y) cannot add new information

13. ROBUSTNESS: MOTIVATION Sn S2 S1 y Symptom Genetic Factors (unobserved) u a x Smoking Cancer • Adding many symptoms does not help. • remains non-identifiable

14. INDEPENDENT: BASED ON DISTINCT SETS OF ASSUMPTION z x y x y z u u a a

15. RELEVANCE: FORMULATION Definition 8Let A be an assumption embodied in model M, and p a parameter in M. A is said to be relevant to p if and only if there exists a set of assumptions S in M such that S and A sustain the identification of p but S alone does not sustain such identification. Theorem 2 An assumption A is relevant to p if and only if A is a member of a minimal set of assumptions sufficient for identifying p.

16. ROBUSTNESS: FORMULATION Definition 5 (Degree of over-identification) A parameter p (of model M) is identified to degree k (read: k-identified) if there are k minimal sets of assumptions each yielding a distinct estimand of p.

17. ROBUSTNESS: FORMULATION b c x y z Minimal assumption sets for c. c c c x x z x z z y y y G3 G1 G2 b Minimal assumption sets for b. x z y

18. FROM MINIMAL ASSUMPTION SETS TO MAXIMAL EDGE SUPERGRAPHS FROM PARAMETERS TO CLAIMS x z e.g., Claim: (Total effect) TE(x,z) = q y x x z z y y TE(x,z)= RzxTE(x,z) = Rzx Rzy ·x Definition A claim C is identified to degree k in model M (graph G), if there are k edge supergraphs of G that permit the identification of C, each yielding a distinct estimand.

19. FROM MINIMAL ASSUMPTION SETS TO MAXIMAL EDGE SUPERGRAPHS FROM PARAMETERS TO CLAIMS x z e.g., Claim: (Total effect) TE(x,z) = q y Nonparametric Definition A claim C is identified to degree k in model M (graph G), if there are k edge supergraphs of G that permit the identification of C, each yielding a distinct estimand. x x z z y y

20. SUMMARY OF ROBUSTNESS RESULTS • Formal definition to ROBUSTNESS of causal claims: • “A claim is robust when it is insensitive to violations of some of the model assumptions relevant to substantiating that claim.” • Graphical criteria and algorithms for computing the degree of robustness of a given causal claim.

21. CONCLUSIONS Structural-model semantics enriched with logic + graphs leads to formal interpretation and practical assessments of wide variety of causal and counterfactual relationships.