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：假设$S$和$\tilde S$是独立同分布的两个随机变量，我们将其解释为信号，并且$P_1$和$P_2$是两个通信信道。我们可以在两个测量场景之间进行选择：要么我们通过$P_1$和$P_2$观察$S$和$\tilde{S}$，要么我们通过$P_1$两次观察$S$，并且通过$P_2$两次观察$\tilde S$。在这两种情况下，哪种情况可以获得最多关于信号$(S, \tilde S)$的信息？虽然当$P_1$和$P_2$为加性高斯信道时第一种情况总是提供更多信息，但我们给出的例子表明这个性质不适用于任意信道。由于这个结果的影响，我们展示了在稀疏随机块模型的社区检测中出现的连续时间互信息不是凸的，即使在大系统规模的情况下也是如此。这与平均度数发散模型的情况形成对比，并给这种数量的渐近行为分析带来了额外的挑战。