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Learn the efficient technique of MAP inference in graphical models using Max-Sum Elimination. This method involves exact inference and product summation to find the maximum a posteriori probability. Understand the process of message passing, convergence, and decoding to decode the MAP assignment accurately. Discover how to handle situations where the MAP assignment may not be unique by perturbing parameters or using a traceback procedure. Gain insights into using the clique tree algorithm for max-sum calculations and decoding the MAP assignment for optimal results.
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Inference Probabilistic Graphical Models MAP Max-Sum Exact Inference
E A B C D Max-Sum Elimination in Chains X
E A B C D Max-Sum Elimination in Chains X X
E A B C D Max-Sum Elimination in Chains X X X X
1: A,B 2: B,C 3: C,D 4: D,E C B D E A B C D Max-Sum in Clique Trees 12(B) = 21(B) = 23(C) = 32(C) = 34(D) = 43(D) =
1: A,B 2: B,C 3: C,D 4: D,E C B D Convergence of Message Passing • Once Ci receives a final message from all neighbors except Cj, then ij is also final (will never change) • Messages from leaves are immediately final 12(B) = 21(B) = 23(C) = 32(C) = 34(D) = 43(D) =
A B C Simple Example
1: A,B 2: B,C B Simple Example
Max-Sum BP at Convergence • Each clique contains max-marginal • Cliques are necessarily calibrated • Agree on sepsets
Decoding a MAP Assignment • Easy if MAP assignment is unique • Single maximizing assignment at each clique • Whose value is the value of the MAP assignment • Due to calibration, choices at all cliques must agree
Decoding a MAP assignment • If MAP assignment is not unique, we may have multiple choices at some cliques • Arbitrary tie-breaking may not produce a MAP assignment
Decoding a MAP assignment • If MAP assignment is not unique, we may have multiple choices at some cliques • Arbitrary tie-breaking may not produce a MAP assignment • Two options: • Slightly perturb parameters to make MAP unique • Use traceback procedure that incrementally builds a MAP assignment, one variable at a time
Summary • The same clique tree algorithm used for sum-product can be used for max-sum • Result is a max-marginal at each clique C: • For each assignment c to C, what is the score of the best completion to c • Decoding the MAP assignment • If MAP assignment is unique, max-marginals allow easy decoding • Otherwise, some additional effort is required