On the relation of nonanticipative rate distortion function and filtering theory

In this paper the relation between nonanticipative rate distortion function (RDF) and Bayesian filtering theory is investigated using the topology of weak convergence of probability measures on Polish spaces. The relation is established via an optimization on the space of conditional distributions of the so-called directed information subject to fidelity constraints. Existence of the optimal reproduction distribution of the nonanticipative RDF is shown, while the optimal nonanticipative reproduction conditional distribution for stationary processes is derived in closed form. The realization procedure of nonanticipative RDF which is equivalent to joint-source channel matching for symbol-by-symbol transmission is described, while an example is introduced to illustrate the concepts.Comment: 6 pages, 4 figures, final version submitted for publication at 12th Biannual European Control Conference (ECC), 201

Optimal Estimation via Nonanticipative Rate Distortion Function and Applications to Time-Varying Gauss-Markov Processes

In this paper, we develop {finite-time horizon} causal filters using the nonanticipative rate distortion theory. We apply the {developed} theory to {design optimal filters for} time-varying multidimensional Gauss-Markov processes, subject to a mean square error fidelity constraint. We show that such filters are equivalent to the design of an optimal \texttt{\{encoder, channel, decoder\}}, which ensures that the error satisfies {a} fidelity constraint. Moreover, we derive a universal lower bound on the mean square error of any estimator of time-varying multidimensional Gauss-Markov processes in terms of conditional mutual information. Unlike classical Kalman filters, the filter developed is characterized by a reverse-waterfilling algorithm, which ensures {that} the fidelity constraint is satisfied. The theoretical results are demonstrated via illustrative examples.Comment: 35 pages, 6 figures, submitted for publication in SIAM Journal on Control and Optimization (SICON

Information Nonanticipative Rate Distortion Function and Its Applications

This paper investigates applications of nonanticipative Rate Distortion Function (RDF) in a) zero-delay Joint Source-Channel Coding (JSCC) design based on average and excess distortion probability, b) in bounding the Optimal Performance Theoretically Attainable (OPTA) by noncausal and causal codes, and computing the Rate Loss (RL) of zero-delay and causal codes with respect to noncausal codes. These applications are described using two running examples, the Binary Symmetric Markov Source with parameter p, (BSMS(p)) and the multidimensional partially observed Gaussian-Markov source. For the multidimensional Gaussian-Markov source with square error distortion, the solution of the nonanticipative RDF is derived, its operational meaning using JSCC design via a noisy coding theorem is shown by providing the optimal encoding-decoding scheme over a vector Gaussian channel, and the RL of causal and zero-delay codes with respect to noncausal codes is computed. For the BSMS(p) with Hamming distortion, the solution of the nonanticipative RDF is derived, the RL of causal codes with respect to noncausal codes is computed, and an uncoded noisy coding theorem based on excess distortion probability is shown. The information nonanticipative RDF is shown to be equivalent to the nonanticipatory epsilon-entropy, which corresponds to the classical RDF with an additional causality or nonanticipative condition imposed on the optimal reproduction conditional distribution.Comment: 34 pages, 12 figures, part of this paper was accepted for publication in IEEE International Symposium on Information Theory (ISIT), 2014 and in book Coordination Control of Distributed Systems of series Lecture Notes in Control and Information Sciences, 201

Applications of Information Nonanticipative Rate Distortion Function

The objective of this paper is to further investigate various applications of information Nonanticipative Rate Distortion Function (NRDF) by discussing two working examples, the Binary Symmetric Markov Source with parameter $p$ (BSMS($p$)) with Hamming distance distortion, and the multidimensional partially observed Gaussian-Markov source. For the BSMS($p$), we give the solution to the NRDF, and we use it to compute the Rate Loss (RL) of causal codes with respect to noncausal codes. For the multidimensional Gaussian-Markov source, we give the solution to the NRDF, we show its operational meaning via joint source-channel matching over a vector of parallel Gaussian channels, and we compute the RL of causal and zero-delay codes with respect to noncausal codes.Comment: 5 pages, 3 figures, accepted for publication in IEEE International Symposium on Information Theory (ISIT) proceedings, 201