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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:. Q 1 : P ( y | do ( x )) Causal Effect (= P ( Y x =y ) ) - PowerPoint PPT Presentation

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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 necessityQ3: Direct Effect

  • Modeling: Statistical vs. CausalCausal Models and IdentifiabilityInference to three types of claims:Effects of potential interventionsClaims about attribution (responsibility)Claims about direct and indirect effectsActual Causation and ExplanationRobustness of Causal Claims

    OUTLINE

  • ROBUSTNESS:MOTIVATIONSmokingxyGenetic Factors (unobserved)Canceru

  • ROBUSTNESS:MOTIVATIONThe claim a = Ryx is sensitive to the assumption cov (x,u) = 0.SmokingxGenetic Factors (unobserved)Cancera is non-identifiable if cov (x,u) 0.yu

  • ROBUSTNESS:MOTIVATIONZ Instrumental variable; cov(z,u) = 0SmokingyGenetic Factors (unobserved)Cancerau

  • ROBUSTNESS:MOTIVATIONSmokingyGenetic Factors (unobserved)CancerauClaim a = Ryx is likely to be true

  • SmokingROBUSTNESS:MOTIVATIONZ1Price ofCigarettesbxyGenetic Factors (unobserved)Cancerau

  • ROBUSTNESS:MOTIVATIONZ1Price ofCigarettesbxyGenetic Factors (unobserved)CancerauPeerPressureZ2gSmoking Greater surprise: a1 = a2 = a3.= an = qClaim a = q is highly likely to be correct

  • ROBUSTNESS:MOTIVATIONAssume we have several independent estimands of a, and

    xyaGiven a parameter a in a general graphFind the degree to which a is robust to violations of model assumptionsa1 = a2 = an

  • ROBUSTNESS:ATTEMPTED FORMULATION Bad attempt: Parameter a is robust (over identifies)

    f1, f2: Two distinct functions

    if:

  • ROBUSTNESS:MOTIVATIONxyGenetic Factors (unobserved)CancerauSmokingIs a robust if a0 = a1?

  • ROBUSTNESS:MOTIVATIONxyGenetic Factors (unobserved)CancerauSmokingSymptoms 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 informationb

  • ROBUSTNESS:MOTIVATIONxGenetic Factors (unobserved)CancerauSmokingAdding many symptoms does not help. remains non-identifiable

  • INDEPENDENT:BASED ON DISTINCT SETS OF ASSUMPTIONauau

    EstimandAssumptioms

    EstimandAssumptioms

  • RELEVANCE:FORMULATIONDefinition 8 Let 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:FORMULATIONDefinition 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

  • FROM MINIMAL ASSUMPTION SETS TO MAXIMAL EDGE SUPERGRAPHS FROM PARAMETERS TO CLAIMSDefinitionA 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 CLAIMSDefinitionA 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.xyzxyz

  • SUMMARY OF ROBUSTNESS RESULTSFormal 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.

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

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