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We propose a general formalism of iterated random functions with semigroupproperty, under which exact and approximate Bayesian posterior updates can beviewed as specific instances. A convergence theory for iterated randomfunctions is presented. As an application of the general theory we analyzeconvergence behaviors of exact and approximate message-passing algorithms thatarise in a sequential change point detection problem formulated via a latentvariable directed graphical model. The sequential inference algorithm and itssupporting theory are illustrated by simulated examples.

Bayesian inference as iterated random functions with applications to sequential inference in graphical models