CSE 571 Advanced Artificial Intelligence Nov 24, 2003 Class Notes Transcribed By: Jon Lammers.

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Presentation transcript:

CSE 571 Advanced Artificial Intelligence Nov 24, 2003 Class Notes Transcribed By: Jon Lammers

10/22/2003CSE Advanced Artificial Intelligence 2 Learning Structured Models Definition – Observational Equivalence – –Two structures are observationally equivalent if the set of data can satisfy both graphs by finding a parameters. Operationally Equivalent

10/22/2003CSE Advanced Artificial Intelligence 3 a is independent of b.P(a/b) = P(a) d is independent of {a,b} given c.P(d/a,b,c) = P(d/c) Consistent/MinimalNot ConsistentConsistent/Minimal Example – Consistency a c d ba c d b * c d b * a

10/22/2003CSE Advanced Artificial Intelligence 4 Example – Coins ABCP(fair)P(-fair) With fair coins, multiple structures can be derived from the data. Including those that do not correctly capture the causal relationship. Making the coins unfair, leaves you with the correct structure. Fair:P(A) = 0.50 P(B) = Fair:P(A) = 0.60 P(B) = 0.30 a c b b a c a b c Correct Structure Only consistent with fair coins.

10/22/2003CSE Advanced Artificial Intelligence 5 Definition Stability – Let I(P) denote the set of all conditional independence relationships embodied in P. A causal model M=(D,Θ D ) generates a stable distribution if and only if P((D,Θ D )) contains no extraneous independencies - that is, if and only if I(P((D,Θ D ))) I(P((D,Θ’ D ))) for any set of parameters Θ’ D.

10/22/2003CSE Advanced Artificial Intelligence 6 Algorithm If you have a large number of nodes, you can have a huge number of possible graphs. The algorithm generates a graph (non-directed) that is a summary of all the minimal stable structures consistent with the data. You can arbitrarily put arrows as long as you do not introduce any new “v-structures”. V-StructureNot V-Structure

10/22/2003CSE Advanced Artificial Intelligence 7 Inductive Causation Algorithm 1.For a, b find S ab such that a independent b/S ab, construct undirected graph with edge between a, b if you can’t find S ab. (not S ab = Ø) 2.For each pair of non-adjacent variables a, b with a common neighbor c. If c is not an element of S ab add arrowheads pointing to c from a and b. 3.Orient as many undirected edges as possible while: –Not creating v-structures –Not creating directed cycles.

10/22/2003CSE Advanced Artificial Intelligence 8 Rules for Orienting Arrows (Step 3) 1.Orient b – c as b -> c when a -> b and a, c are non- adjacent. 2.Orient a – b as a -> b when a -> c -> b. 3.Orient a – b as a -> b when a – c -> b and a – d -> b and b, c are non-adjacent. 4.Orient a – b as a -> b when a – c -> d and c -> d -> b and c, b are non-adjacent.