This paper estimates free energy, average mutual information, and minimum mean square error (MMSE) of a linear model under two assumptions: (1) the source is generated by a Markov chain, (2) the source is generated via a hidden Markov model. Our estimates are based on the replica method in statistical physics. We show that under the posterior mean estimator, the linear model with Markov sources or hidden Markov sources is decoupled into single-input AWGN channels with state information available at both encoder and decoder where the state distribution follows the left Perron-Frobenius eigenvector with unit Manhattan norm of the stochastic matrix of Markov chains. Numerical results show that the free energies and MSEs obtained via the replica method are closely approximate to their counterparts achieved by the Metropolis-Hastings algorithm or some well-known approximate message passing algorithms in the research literature.

翻译：本文在线性模型下估计了自由能、平均互信息以及最小均方误差（MMSE），并基于两种假设：（1）信源由马尔可夫链生成，（2）信源通过隐马尔可夫模型生成。我们的估计基于统计物理学中的副本方法。研究表明，在后验均值估计器下，带有马尔可夫信源或隐马尔可夫信源的线性模型可解耦为多个单输入加性高斯白噪声（AWGN）信道，其中编码器和解码器均可获取状态信息，且状态分布遵循马尔可夫链随机矩阵的左佩龙-弗罗贝尼乌斯特征向量，该向量具有单位曼哈顿范数。数值结果表明，通过副本方法获得的自由能和均方误差（MSE）与文献中通过梅特罗波利斯-黑斯廷斯算法或一些著名的近似消息传递算法所得到的结果高度吻合。