Judea Pearl Computer Science Department UCLA ROBUSTNESS OF CAUSAL CLAIMS

ROBUSTNESS: MOTIVATION The effect of smoking on cancer is, in general, non-identifiable (from observational studies). Smoking x y Genetic Factors (unobserved) Cancer u In linear systems: y =  x +   is non-identifiable. 

ROBUSTNESS: MOTIVATION Z – Instrumental variable; cov( z,u ) = 0 Smoking y Genetic Factors (unobserved) Cancer  u x Z Price of Cigarettes   is identifiable

ROBUSTNESS: MOTIVATION Problem with Instrumental Variables: The model may be wrong! Smoking Z Price of Cigarettes  x y Genetic Factors (unobserved) Cancer  u

Smoking ROBUSTNESS: MOTIVATION Z1Z1 Price of Cigarettes  Solution: Invoke several instruments Surprise:  1 =  2 model is likely correct x y Genetic Factors (unobserved) Cancer  u Peer Pressure Z2Z2 

ROBUSTNESS: MOTIVATION Z1Z1 Price of Cigarettes  x y Genetic Factors (unobserved) Cancer  u Peer Pressure Z2Z2  Smoking Greater surprise:  1 =  2 =  3 ….=  n = q Claim  = q is highly likely to be correct Z3Z3 ZnZn Anti-smoking Legislation

ROBUSTNESS: MOTIVATION xy Genetic Factors (unobserved) Cancer  u Smoking Symptoms do not act as instruments  remains non-identifiable s Symptom Why? Taking a noisy measurement ( s ) of an observed variable ( y ) cannot add new information

ROBUSTNESS: MOTIVATION x Genetic Factors (unobserved) Cancer  u Smoking Adding many symptoms does not help.  remains non-identifiable y Symptom S1S1 S2S2 SnSn

ROBUSTNESS: MOTIVATION Find if  can evoke an equality surprise  1 =  2 = …  n associated with several independent estimands of  x y  Given a parameter  in a general graph Formulate: Surprise, over-identification, independence Robustness: The degree to which  is robust to violations of model assumptions

ROBUSTNESS: FORMULATION Bad attempt: Parameter  is robust (over identifies) f 1, f 2 : Two distinct functions if:

ROBUSTNESS: FORMULATION exex eyey ezez xyz bc x = e x y = bx + e y z = cy + e z R yx = b R zx = bc R zy = c constraint: (b) (c) y → z irrelvant to derivation of b

RELEVANCE: FORMULATION Definition 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: 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 xy b z c Minimal assumption sets for c. x y z c x y z c G3G3 G2G2 x y z c G1G1 Minimal assumption sets for b. x y b z

FROM MINIMAL ASSUMPTION SETS TO MAXIMAL EDGE SUPERGRAPHS FROM PARAMETERS TO CLAIMS 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. TE ( x,z ) = R zx TE ( x,z ) = R zx Rzy ·x x y z x y z e.g., Claim: (Total effect) TE (x,z) = q x y z

CONCLUSIONS 1.Formal definition to ROBUSTNESS of causal claims: “A claim is robust when it is insensitive to violations of some of the model assumptions” 2.Graphical criteria and algorithms for computing the degree of robustness of a given causal claim.