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 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：本文根据以下两个假设估计线性模型的免费能源、平均相互信息和最小平均正方差:(1) 源由Markov链生成,(2) 源由隐藏的Markov模型生成,我们的估计数以统计物理的复制方法为基础。我们显示,在后方平均估计值下,带有Markov源或隐藏的Markov源的线性模型被分解成单输入的AWGN频道,在编码器和解码器上都有国家信息,在编码器和解码器上,国家分布都遵循左端的Perron-Frobenius 元子以及Markov链的随机矩阵曼哈顿单位规范。数字结果显示,通过复制方法获得的自由能量和MSE与Metropolis-Hastings算法或研究文献中一些众所周知的近似电文传动算法所实现的对应方十分接近。