matrix, $\y\in\mathbb{R}^{MJ}$ are the measurements, and the noise is $\z\in\mathbb{R}^{MJ}$. Entries of the signal vectors $\underline{\s}^j$, measurement vectors $\underline{\y}^j$, and noise vectors $\underline{\z}^j$  form the SMV signal $\s$, measurements $\y$, and noise $\z$  with $$s_{(l-1)J+j}=\underline{s}^{j}_l,\ y_{(j-1)M+\mu}=\underline{y}^j_{\mu},\ \text{and}\ z_{(j-1)M+\mu}=\underline{z}^j_{\mu},$$ respectively. Entries of the matrix $\underline{\F}^j$  form the SMV matrix $\F$  with $F_{(j-1)M+\mu,(l-1)J+j}=\underline{F}^j_{\mu l}$; other entries of $\F$ are zeros. The posterior for the estimate $\x\in\mathbb{R}^{NJ}$, comprised of super symbols $\x_l=[x_{(l-1)J+1},...,x_{lJ}]^T,\ l\in\{1,...,N\}$, is $$\label{eq:pxy2}
  P(\x|\y)=\frac{1}{Z}{\prod_{l=1}^{N}}P(\x_l){\prod_{\mu=1}^{MJ}}\l[\frac{\operatorname{e}^{-\frac{1}{2\Delta}(\\y_{\mu}-{\sum_{l=1}^{N}}\F_{\mu l}\x_l)^2}}{\sqrt{2\pi\Delta}}\r],$$ where $\F_{\mu l}=[F_{\mu,(l-1)J+1}, \ldots, F_{\mu,lJ}]$ is a super symbol highlighted by the dashed area in Fig. \[fig:channel\], and the denominator $Z$ is the partition function [@Tanaka2002; @GuoVerdu2005; @Krzakala2012probabilistic; @krzakala2012statistical; @MezardMontanariBook; @Barbier2015], $$\label{eq:partition}
  Z=\int {\prod_{l=1}^{N}}P(\x_l){\prod_{\mu=1}^{MJ}}\l[\frac{\operatorname{e}^{-\frac{1}{2\Delta}(y_{\mu}-{\sum_{l=1}^{N}}\F_{\mu l}\x_l)^2}}{\sqrt{2\pi\Delta}}\r]{\prod_{l=1}^{N}}d\x_l.$$ Note that multi-dimensional integrations such as  are denoted by a single $\int$ operator for brevity. Confining our attention to the Bayesian setting [@Krzakala2012probabilistic; @krzakala2012statistical; @Barbier2015], $P(\x_l)$ follows the true distribution , $P(\x_l)=\rho \phi(\x_l)+(1-\rho)\delta(\x_l)$.

By creating an analogy between the channel  and a many-body thermodynamic system [@Tanaka2002; @GuoVerdu2005; @Krzakala2012probabilistic; @krzakala2012statistical; @MezardMontanariBook; @Barbier2015], the posterior  can be interpreted as the Boltzmann measure on a disordered system with the following Hamiltonian, $$\label{eq:Hamiltonian}
H(\x)=\sum_{l=1}^N \log [P(\x_l)]+\sum_{\mu=1}^{MJ} \frac{1}{2\Delta}\l(y_{\mu}-\sum_{l=1}^N\F_{\mu l} \x_l\r)^2.$$

The averaged free energy of the disordered system given by  characterizes the thermodynamic properties of the 