Realizable Rate Distortion Function and Bayesian FIltering Theory

TL;DR

This paper explores the connection between rate distortion functions and Bayesian filtering, introducing a causal RDF with recursive solutions and demonstrating its realization through source-channel matching.

Contribution

It defines a causal rate distortion function with existence and explicit recursive solutions, linking it to Bayesian filtering and source-channel matching.

Findings

Existence of optimal causal reconstruction distribution proven.

Closed-form recursive equations derived for non-stationary causal RDF.

Causal RDF realization via source-channel matching demonstrated.

Abstract

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.