Breakdown of a concavity property of mutual information for non-Gaussian channels
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Let S and S̃ be two independent and identically distributed random variables, which we interpret as the signal, and let P_1 and P_2 be two communication channels. We can choose between two measurement scenarios: either we observe S through P_1 and P_2, and also S̃ through P_1 and P_2; or we observe S twice through P_1, and S̃ twice through P_2. In which of these two scenarios do we obtain the most information on the signal (S, S̃)? While the first scenario always yields more information when P_1 and P_2 are additive Gaussian channels, we give examples showing that this property does not extend to arbitrary channels. As a consequence of this result, we show that the continuous-time mutual information arising in the setting of community detection on sparse stochastic block models is not concave, even in the limit of large system size. This stands in contrast to the case of models with diverging average degree, and brings additional challenges to the analysis of the asymptotic behavior of this quantity.
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