Replica Analysis of the Linear Model with Markov or Hidden Markov Signal Priors

TL;DR

This paper uses the replica method to analyze the free energy, mutual information, and MMSE of linear models with Markov or hidden Markov sources, revealing their decoupling into single-input channels.

Contribution

It introduces a novel application of the replica method to linear models with Markov and hidden Markov priors, providing analytical estimates of key information-theoretic quantities.

Findings

Replica estimates closely match Metropolis-Hastings results.

Decoupling into single-input AWGN channels with state information.

Analytical expressions for free energy and MMSE.

Abstract

This paper estimates free energy, average mutual information, and minimum mean square error (MMSE) of a linear model under two assumptions: (1) the source is generated by a Markov chain, (2) the source is generated via a hidden Markov model. Our estimates are based on the replica method in statistical physics. We show that under the posterior mean estimator, the linear model with Markov sources or hidden Markov sources is decoupled into single-input AWGN channels with state information available at both encoder and decoder where the state distribution follows the left Perron-Frobenius eigenvector with unit Manhattan norm of the stochastic matrix of Markov chains. Numerical results show that the free energies and MSEs obtained via the replica method are closely approximate to their counterparts achieved by the Metropolis-Hastings algorithm or some well-known approximate message passing algorithms in the research literature.