Statistical Physics Analysis of Maximum a Posteriori Estimation for Multi-channel Hidden Markov Models

TL;DR

This paper analyzes MAP estimation in multi-channel Hidden Markov Models using statistical physics, revealing phase transitions and solution multiplicity, and introduces a semi-analytical method to evaluate estimation errors.

Contribution

It maps the MAP estimation problem to a 1D Ising model, characterizes phase transitions, and develops a semi-analytical approach for error calculation in multi-channel HMMs.

Findings

Operational regimes separated by first order phase transitions.

Number of solutions depends on odd/even number of channels.

Semi-analytical method for estimation error calculation.

Abstract

The performance of Maximum a posteriori (MAP) estimation is studied analytically for binary symmetric multi-channel Hidden Markov processes. We reduce the estimation problem to a 1D Ising spin model and define order parameters that correspond to different characteristics of the MAP-estimated sequence. The solution to the MAP estimation problem has different operational regimes separated by first order phase transitions. The transition points for $L$-channel system with identical noise levels, are uniquely determined by $L$ being odd or even, irrespective of the actual number of channels. We demonstrate that for lower noise intensities, the number of solutions is uniquely determined for odd $L$, whereas for even $L$ there are exponentially many solutions. We also develop a semi analytical approach to calculate the estimation error without resorting to brute force simulations. Finally, we examine the tradeoff between a system with single low-noise channel and one with multiple noisy channels.