all the information needed to update the estimate.

The optimal estimate of each super-symbol $\x_n$ in the signal vectors is $$\label{eq:estimatorTrue}
\widetilde{\x}_{\text{true},n}=\arg\min_{\widehat{\x}_n} \int d_{\text{UD}}(\x_n,\widehat{\x}_n) f(\x_n|\{\y^{(j)}\}_{j=1}^J) d\x_n.$$ Comparing  and , provided that $f(\x_n|\q_n)$ converges to the posterior $f(\x_n|\{\y^{(j)}\}_{j=1}^J)$, we have $$\widetilde{\x}_n=\arg\min_{\widehat{\x}_n} \mathbb{E}[d_{\text{UD}}(\x_n,\widehat{\x}_n)|\q_n]\approx \widetilde{\x}_{\text{true},n},$$ which results in Conjecture \[conj:2\].

Example Metric-optimal Estimators and Performance Limits {#sec:example}
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In order to derive metric-optimal estimators, we need to know the scalar channel noise variance, $\Delta_v$. Below we obtain the optimal estimator, which is then used as part of Algorithm \[algo:metric\_opt\_MMV\] and to evaluate the performance limits of our metric-optimal algorithm.

For i.i.d. matrices and AWGN channels , replica analysis in our previous work with Krzakala [@ZhuBaronKrzakala2017IEEE] yields the information-theoretic scalar channel noise variance $\Delta_v$ for message passing algorithms, which characterizes the posterior $f(\x_n|\q_n)$.[^8] Hence, we can obtain the information-theoretic optimal performance with arbitrary additive error metrics. For other types of matrices, our replica analysis [@ZhuBaronKrzakala2017IEEE] does not hold. Nevertheless, MP algorithms still yield a posterior $f(\x_n|\q_n)$, which we conjecture converges to the true posterior $f(\x_n|\{\y^{(j)}\}_{j=1}^J)$ (Conjecture \[conj:2\]). Hence, we can still assume that the scalar channel noise variance $\Delta_v$ is known. Based on the known variance $\Delta_v$, we build metric-optimal estimators  for two examples, 