On Maximum a Posteriori Estimation of Hidden Markov Processes

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.

💡 Research Summary

The paper investigates the maximum‑a‑posteriori (MAP) sequence estimation problem for binary symmetric hidden Markov processes (HMPs) by mapping it onto the ground‑state problem of a one‑dimensional Ising spin chain with nearest‑neighbour coupling and a random external field. The hidden states (x_i\in{+1,-1}) evolve according to a two‑state Markov chain with transition probability (p), while each observation (y_i) is obtained by passing (x_i) through a symmetric binary channel that flips the bit with probability (\varepsilon). The MAP estimator seeks the state sequence that maximizes the posterior probability (P(\mathbf{x}\mid\mathbf{y})), which is equivalent to minimizing the negative log‑posterior. By introducing spin variables (s_i) for the hidden states, the authors rewrite the log‑posterior as an Ising Hamiltonian

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