  We propose a general formalism of iterated random functions with semigroup
property, under which exact and approximate Bayesian posterior updates can be
viewed as specific instances. A convergence theory for iterated random
functions is presented. As an application of the general theory we analyze
convergence behaviors of exact and approximate message-passing algorithms that
arise in a sequential change point detection problem formulated via a latent
variable directed graphical model. The sequential inference algorithm and its
supporting theory are illustrated by simulated examples.
