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

摘要

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.