  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.
