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We explore the fundamental limits of heterogeneous distributed detection inan anonymous sensor network with n sensors and a single fusion center. Thefusion center collects the single observation from each of the n sensors todetect a binary parameter. The sensors are clustered into multiple groups, anddifferent groups follow different distributions under a given hypothesis. Thekey challenge for the fusion center is the anonymity of sensors - although itknows the exact number of sensors and the distribution of observations in eachgroup, it does not know which group each sensor belongs to. It is hence naturalto consider it as a composite hypothesis testing problem. First, we propose anoptimal test called mixture likelihood ratio test based on the ratio of theuniform mixture of all distributions under one hypothesis to that under theother. Optimality is shown by first arguing that symmetric tests are optimal,which do not depend on the order of observations across the sensors, and thenproving that the mixture likelihood ratio test is optimal among all symmetrictests. Second, we focus on the Neyman-Pearson setting and characterize theerror exponent of the worst-case type-II error probability as n tends toinfinity, assuming the number of sensors in each group is proportional to n.Finally, we generalize our result to find the collection of all achievabletype-I and type-II error exponents, where the boundary of the region can beobtained by solving a convex program. Our results elucidate the price ofanonymity in heterogeneous distributed detection, and can be extended to M-aryhypothesis testing with heterogeneous observations generated according tohidden latent variables. The results are also applied to distributed detectionunder Byzantine attacks, which hints that the conventional approach based onsimple hypothesis testing might be too pessimistic.

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