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Explore the robustness of causal claims in structural models, examining the sensitivity to violations of model assumptions and relevance to substantiating claims. Learn about inferential approaches and graphical criteria for assessing claim validity.
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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
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
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
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.
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
ROBUSTNESS: MOTIVATION Z Price of Cigarettes b x Genetic Factors (unobserved) u a y Smoking Cancer Claim “a = Ryx”is likely to be true
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
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
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
ROBUSTNESS: ATTEMPTED FORMULATION Bad attempt: Parameter a is robust (over identifies) f1, f2: Two distinct functions if:
ROBUSTNESS: MOTIVATION b s Symptom Genetic Factors (unobserved) u a x y Smoking Cancer Is a robust if a0 = a1?
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
ROBUSTNESS: MOTIVATION Sn S2 S1 y Symptom Genetic Factors (unobserved) u a x Smoking Cancer • Adding many symptoms does not help. • remains non-identifiable
INDEPENDENT: BASED ON DISTINCT SETS OF ASSUMPTION z x y x y z u u a a
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.
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.
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
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.
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
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.
CONCLUSIONS Structural-model semantics enriched with logic + graphs leads to formal interpretation and practical assessments of wide variety of causal and counterfactual relationships.