free energy expression for a CS problem, the MMSE can be obtained by evaluating the largest free energy [@Tanaka2002; @GuoVerdu2005; @Krzakala2012probabilistic; @krzakala2012statistical; @MezardMontanariBook; @Barbier2015]. Having derived the free energy for the two MMV settings in Section \[sec:main\], this section calculates the MMSE under various cases. Different performance regions of MMV are identified, where the MMSE behaves differently as a function of the noise variance $\Delta$ and measurement rate $R$. We identify a phase transition of belief propagation (BP) that separates regions where BP is optimal asymptotically or not. Simulation results match the predicted performance of BP.

Performance regions: Definitions and numerical results {#sec:PerfRegion}
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When calculating the MMSE  for different settings from the free energy expression , four different [*performance regions*]{} will appear, as illustrated in Fig. \[fig:freeEnergyProf\] and discussed below.

![Free energy as a function of MSE for different measurement rates $R$ (number of jointly sparse signal vectors $J=3$ and noise variance $\Delta=-35$ dB). The black circles mark the largest free energy, and so they correspond to the MMSE.[]{data-label="fig:freeEnergyProf"}](freeEnergyProfiles_J3.pdf){width="8cm"}

[**Regions 1 and 4:**]{} The free energy  has one local maximum point w.r.t. the MSE $D$ . This $D$ leads to the globally maximum free energy and is the MMSE.

