THEORY AND APPLICATIONS OF CAUSAL REASONING CONGNITIVE SYSTEMS LABORATORY UCLA Judea Pearl and Jin Tian Model Correctness Mark Hopkins LAYER WIDTH: A New Measure of DAG’s Size Blai Bonet Labeled RTDP: A Fast Dynamic Programming Algorithm Carlos Brito Graphical Methods for Identifying SEM

STRUCTURES, DATA, AND CORRECTNESS OBSERVATIONAL DATA ( P(A,B,C,D)) ABCD YESNOHIGHYES NOYESHIGHYES NO LOWNO … EXPERIMENTAL DATA ( P B=yes (A,B,C,D)) ABCD YES LOWYES NOYESHIGHNO YESLOWYES … ABC U D Genetic factor Cancer Tar in LungsSmoking Stress U is unobservable (unobserved)

MODEL CORRECTNESS Correctness: 1. 1.The structure has empirical implications 2. 2.Empirical data complies with those implications 3. 3.The structure given is the only one to satisfy 1-2. If 1 and 2 are satisfied, we say that the data “corroborates” the structure. Empirical implications may be observational or experimental. Given a structure of a model, in what sense can we assert that the structure is “correct”?

STRUCTURES AND IMPLICATIONS CONSIDERED ObservationalExperimental Specific Markovian Specific Semi Markovian Generic Markovian Generic Semi Markovian CI d -separation, complete Pearl, 2000 (Complete) Tian & Pearl (Complete) Algorithmic IC, Sprites et al. (Complete) Tian & Pearl Incomplete? Algorithmic IC*, Sprites et al. (Incomplete?) CI + functional (Tian & Pearl) Q -decomposition (Tian & Pearl)

MODEL CORRECTNESS Given a structure of a model, in what sense can we assert that the structure is “correct”? Correctness: 1. 1.The structure has empirical implications Empiricial data complies with those implications The structure given is the only one to satisfy 1-2. If 1 and 2 are satisfied, we say that the data “corroborates” the structure. Empiricial implications may be observational or experimental. CLAIM a claim implied by Need to define: 1.Substructure essential for claim 2.Data corroborates claim

a = r YX xyxy a StructureClaim e.g., FROM CORROBORATING MODELS TO CORROBORATING CLAIMS Is the claim a = r YX corroborated? No! Because the assumption r s = 0 is essential for the claim and no data can corroborate this assumption. xyxy a rsrs a = r YX - r s

FROM CORROBORATING MODELS TO CORROBORATING CLAIMS e.g., StructureClaim b = r ZY Is the claim b = r ZY corroborated? YES! Because the assumption needed for entailing this claim, r b = 0, can be tested: b xyzxy a b xy z xy a r ZX = r YX r ZY rbrb Note: Assumption r a = 0 cannot be tested. rara

WHEN IS AN ASSUMPTION NEEDED? Definition (Relevance): Assumption A is relevant to claim C iff there exists a set S of assumptions in the model such that Definition: A claim C is corroborated by data iff the sum total of all assumptions relevant to C is corroborated by data.

a GRAPHICAL CRITERION FOR CORROBORATED CLAIMS Theorem: An identifiable claim C is corroborated by data if The intersection of all maximal supergraphs sufficient for identifying C is corroborated by the data. b a xyz b e.g., xyx a xyz Maximal supergraph for claim a = a 0

GRAPHICAL CRITERION FOR CORROBORATED CLAIMS Theorem: An identifiable claim C is corroborated by data if The intersection of all maximal supergraphs sufficient for identifying C is corroborated by the data. e.g., b a xyzxyx b a xyz ab a xyzxyx b a xyzxyx b a xyz Intersection: xyx Maximal supergraphs:

SOME CLAIMS ARE MORE CORROBORATED THAN OTHERS G ( C ) = Intersection of all maximal supergraphs sufficient for identifying C. Definition: Claim C 1 is more corroborated than claim C 2 if the constraints induced by G ( C 1 ) entail those induced by G ( C 2 ).

The strongest sense in which one can proclaim a model "correct" is that the data comply with the observational or experimental implication of the model. Jin has explicated those implications for both Markovian and Semi-Markovian structures. I have extended these considerations from models to claims, and obtained graphical criteria for determining in what sense a specific claim C can be proclaimed "correct". CONCLUSIONS