Anonymous Heterogeneous Distributed Detection: Optimal Decision Rules, Error Exponents, and the Price of Anonymity

We explore the fundamental limits of heterogeneous distributed detection in an anonymous sensor network with n sensors and a single fusion center. The fusion center collects the single observation from each of the n sensors to detect a binary parameter. The sensors are clustered into multiple groups, and different groups follow different distributions under a given hypothesis. The key challenge for the fusion center is the anonymity of sensors - although it knows the exact number of sensors and the distribution of observations in each group, it does not know which group each sensor belongs to. It is hence natural to consider it as a composite hypothesis testing problem. First, we propose an optimal test called mixture likelihood ratio test based on the ratio of the uniform mixture of all distributions under one hypothesis to that under the other. Optimality is shown by first arguing that symmetric tests are optimal, which do not depend on the order of observations across the sensors, and then proving that the mixture likelihood ratio test is optimal among all symmetric tests. Second, we focus on the Neyman-Pearson setting and characterize the error exponent of the worst-case type-II error probability as n tends to infinity, assuming the number of sensors in each group is proportional to n. Finally, we generalize our result to find the collection of all achievable type-I and type-II error exponents, where the boundary of the region can be obtained by solving a convex program. Our results elucidate the price of anonymity in heterogeneous distributed detection, and can be extended to M-ary hypothesis testing with heterogeneous observations generated according to hidden latent variables. The results are also applied to distributed detection under Byzantine attacks, which hints that the conventional approach based on simple hypothesis testing might be too pessimistic.