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Properties of Belief Propagation Algorithm in Probabilistic Graphical Models

This text explores the properties of the Belief Propagation (BP) algorithm in probabilistic graphical models, focusing on the concepts of calibration and convergence. It describes how a cluster graph is calibrated when adjacent clusters agree on their sepsets. The discussion includes sepset marginals and their implications for reparameterization. It emphasizes that at convergence, the beliefs at adjacent clusters align, resulting in a calibrated parameterization of the original unnormalized density, ensuring that no information is lost during message passing.

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Properties of Belief Propagation Algorithm in Probabilistic Graphical Models

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  1. Inference Probabilistic Graphical Models Message Passing Properties of BP Algorithm

  2. Calibration • Cluster beliefs: • A cluster graph is calibrated if every pair of adjacent clusters Ci,Cjagree on their sepsetSi,j 1: A,B 4: A,D A B D 2: B,C 3: C,D C

  3. Convergence  Calibration • Convergence:

  4. Reparameterization • Sepsetmarginals: 1: A,B 4: A,D A B D 2: B,C 3: C,D C

  5. Reparameterization • Sepsetmarginals:

  6. Summary • At convergence of BP, cluster graph beliefs are calibrated: • beliefs at adjacent clusters agree on sepsets • Cluster graph beliefs are an alternative, calibrated parameterization of the original unnormalized density • No information is lost by message passing

  7. END END END

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