We estimate the minimum mean square error (MMSE) of the linear model under hidden Markov priors. Our estimates are based on the replica method in statistical physics. We show that under the MMSE estimator, the linear model with 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.