THEORY AND APPLICATIONS OF CAUSAL REASONING CONGNITIVE SYSTEMS LABORATORY UCLA

1. 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

2. STRUCTURES, DATA, AND CORRECTNESS Genetic factor U (unobserved) Stress A B C D Cancer Smoking Tar in Lungs U is unobservable

3. MODEL CORRECTNESS Given a structure of a model, in what sense can we assert that the structure is “correct”? Correctness: • The structure has empirical implications • Empirical 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. Empirical implications may be observational or experimental.

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

5. CLAIM a claim implied by MODEL CORRECTNESS Given a structure of a model, in what sense can we assert that the structure is “correct”? Correctness: • 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. • Need to define: • Substructure essential for claim • Data corroborates claim

6. Structure Claim e.g., a = rYX x x y y a Is the claim a = rYX corroborated? No! Because the assumption rs = 0 is essential for the claim and no data can corroborate this assumption. rs a = rYX- rs x x y y a FROM CORROBORATING MODELS TO CORROBORATING CLAIMS

7. Structure Claim x x y y z a b ra rb FROM CORROBORATING MODELS TO CORROBORATING CLAIMS e.g., b = rZY Is the claim b = rZY corroborated? YES! Because the assumption needed for entailing this claim, rb= 0, can be tested: rZX = rYX rZY z x x y y a b Note: Assumption ra = 0 cannot be tested.

8. 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.

9. a e.g., x x x y y z x y z a a b b Maximal supergraph for claima = a0 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.

10. x x x y y z a b Maximal supergraphs: x x x y y z x x x y y z x y z a a b b a a b Intersection: x x x y y z a b 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.,

11. SOME CLAIMS ARE MORE CORROBORATED THAN OTHERS G(C) = Intersection of all maximal supergraphs sufficient for identifying C. Definition: Claim C1 is more corroborated than claim C2 if the constraints induced by G(C1) entail those induced by G(C2).

12. CONCLUSIONS 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".