= Solution
Let $c_s=\widetilde c_s-E_s$ and $M_s=\widetilde M_s-RE_s$. With the stated <improper prior>[improper hyperpriors], the unnormalized <posterior density> is
$$
\begin{aligned}
&p(\{E_s\},M_0,c_0,\beta,\sigma_M^2,\sigma_c^2,R,\tau\mid\{d_s\})\\
&\quad\propto
\mathbf1_{\{\sigma_M^2,\sigma_c^2,\tau>0\}}
\prod_{s=1}^N
\left[
\frac1{\sigma_c}\exp\!\left\{-\frac{(\widetilde c_s-E_s-c_0)^2}{2\sigma_c^2}\right\}
\frac1{\sigma_M}\exp\!\left\{-\frac{[\widetilde M_s-RE_s-M_0-\beta(\widetilde c_s-E_s)]^2}{2\sigma_M^2}\right\}
\frac1\tau e^{-E_s/\tau}\mathbf1_{\{E_s\geq0\}}
\right].
\end{aligned}
$$
Its <Directed acyclic graph> has, for each star, the arrows
$$
(c_0,\sigma_c^2)\to c_s,
\qquad
(M_0,\beta,\sigma_M^2,c_s)\to M_s,
\qquad
\tau\to E_s,
$$
followed by the deterministic observation arrows
$$
(c_s,E_s)\to\widetilde c_s,
\qquad
(M_s,E_s,R)\to\widetilde M_s.
$$
The <latent variables> $(c_s,M_s,E_s)$ are repeated inside a plate indexed by $s=1,\ldots,N$; all hyperparameters lie outside that plate.
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