On Maximum a Posteriori Estimation of Hidden Markov Processes

TL;DR

This paper analyzes MAP estimation for binary symmetric hidden Markov processes by mapping it to an Ising spin model, revealing phase transitions and the relationship between noise levels, accuracy, and solution multiplicity.

Contribution

It provides a theoretical framework connecting MAP estimation performance to thermodynamic phases of an Ising model, highlighting phase transitions and solution multiplicity.

Findings

Unique solutions at low noise levels

Nearly noise-independent accuracy at intermediate noise

Exponential solution multiplicity with non-zero entropy

Abstract

We present a theoretical analysis of Maximum a Posteriori (MAP) sequence estimation for binary symmetric hidden Markov processes. We reduce the MAP estimation to the energy minimization of an appropriately defined Ising spin model, and focus on the performance of MAP as characterized by its accuracy and the number of solutions corresponding to a typical observed sequence. It is shown that for a finite range of sufficiently low noise levels, the solution is uniquely related to the observed sequence, while the accuracy degrades linearly with increasing the noise strength. For intermediate noise values, the accuracy is nearly noise-independent, but now there are exponentially many solutions to the estimation problem, which is reflected in non-zero ground-state entropy for the Ising model. Finally, for even larger noise intensities, the number of solutions reduces again, but the accuracy is poor. It is shown that these regimes are different thermodynamic phases of the Ising model that are related to each other via first-order phase transitions.