The problem of choosing among candidate wireless sensor networks (WSNs) that send data to a fusion center to detect a binary phenomenon in nature is considered. Viewing the system as a communication channel in which nature is the transmitter and the fusion center is the receiver, it is intuitive to prefer the system that provides the highest mutual information between the phenomenon and the received signal at the fusion center. This paper reviews existing literature on this criterion and provides complementary results showing that, under a Bayesian framework, a WSN that provides equal or higher mutual information than other WSNs results in equal or better detection performance only in very restrictive settings. This paper also presents a bound on the detection performance of a WSN having the highest mutual information, and it is shown that the bound becomes tight as the mutual information approaches its maximum. Similar conclusions are obtained in the Neyman-Pearson framework.