Let $S$ and $\tilde S$ 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 $\tilde S$ through $P_1$ and $P_2$; or we observe $S$ twice through $P_1$, and $\tilde{S}$ twice through $P_2$. In which of these two scenarios do we obtain the most information on the signal $(S, \tilde S)$? 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.

翻译：令 $S$ 和 $\tilde S$ 为两个独立同分布的随机变量（将其视为信号），$P_1$ 和 $P_2$ 为两个通信信道。我们可在以下两种测量方案中选择：要么通过 $P_1$ 和 $P_2$ 观测 $S$，同时通过 $P_1$ 和 $P_2$ 观测 $\tilde S$；要么通过 $P_1$ 两次观测 $S$，通过 $P_2$ 两次观测 $\tilde S$。在这两种方案中，哪一种能获得关于信号 $(S, \tilde S)$ 的最大信息量？尽管当 $P_1$ 和 $P_2$ 为加性高斯信道时，第一种方案总能获得更多信息，但我们通过实例表明，该性质并不适用于任意信道。基于此结果，我们进一步证明：在稀疏随机块模型的社区检测场景中，连续时间互信息即使在系统规模趋于无穷的极限下也不具有凹性。这与平均度发散模型的情况形成鲜明对比，并为分析该量的渐近行为带来了额外挑战。