  The relation between rate distortion function (RDF) and Bayesian filtering
theory is discussed. The relation is established by imposing a causal or
realizability constraint on the reconstruction conditional distribution of the
RDF, leading to the definition of a causal RDF. Existence of the optimal
reconstruction distribution of the causal RDF is shown using the topology of
weak convergence of probability measures. The optimal non-stationary causal
reproduction conditional distribution of the causal RDF is derived in closed
form; it is given by a set of recursive equations which are computed backward
in time. The realization of causal RDF is described via the source-channel
matching approach, while an example is briefly discussed to illustrate the
concepts.
