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In this paper the relation between nonanticipative rate distortion function(RDF) and Bayesian filtering theory is investigated using the topology of weakconvergence of probability measures on Polish spaces. The relation isestablished via an optimization on the space of conditional distributions ofthe so-called directed information subject to fidelity constraints. Existenceof the optimal reproduction distribution of the nonanticipative RDF is shown,while the optimal nonanticipative reproduction conditional distribution forstationary processes is derived in closed form. The realization procedure ofnonanticipative RDF which is equivalent to joint-source channel matching forsymbol-by-symbol transmission is described, while an example is introduced toillustrate the concepts.

On the relation of nonanticipative rate distortion function and filtering theory