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.

本文探讨了使用 Ising 自旋模型进行二进制对称隐马尔可夫过程的最大后验(MAP)序列估计的理论分析。我们证明了这个估计问题可以被转化为能量最小化问题，并研究了MAP的性能特征包括准确性和对于观测序列的解的数量。我们的研究结果表明，在适当的噪音强度范围内，求解与观测序列唯一相关，但准确性随噪音增强线性降低；噪声值适中时，准确性几乎独立于噪声，但问题存在指数级解的数量；对于更大的噪声强度，解的数量再次减少，但准确性很差。我们证明这些不同噪声强度范围的Ising模型状态属于不同热力学相，相互相关，并通过一阶相变实现转换。

隐马尔可夫过程的最大后验估计