Replica Analysis of the Linear Model with Markov or Hidden Markov Signal Priors
This work provides theoretical estimates for signal processing and statistical inference problems, but it is incremental as it extends existing replica method analyses to Markov and hidden Markov priors.
The paper tackled the problem of estimating free energy, mutual information, and minimum mean square error for linear models with Markov or hidden Markov signal priors, using the replica method from statistical physics, and showed that numerical results closely approximate those from Metropolis-Hastings or approximate message passing algorithms.
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.