Optimal grouping for group minimax hypothesis testing

Abstract

Bayesian hypothesis testing and minimax hypothesis testing represent extreme instances of detection in which the prior probabilities of the hypotheses are either completely and precisely known, or are completely unknown. Group minimax, also known as \(\Gamma \) -minimax, is a robust intermediary between Bayesian and minimax hypothesis testing that allows for coarse or partial advance knowledge of the hypothesis priors by using information on sets in which the prior lies. Existing work on group minimax, however, does not consider the question of how to define the sets or groups of priors; it is assumed that the groups are given. In this paper, we propose a novel intermediate detection scheme formulated through the quantization of the space of prior probabilities that optimally determines groups and also representative priors within the groups. We show that when viewed from a quantization perspective, group minimax amounts to determining centroids with a minimax Bayes risk error divergence distortion criterion: the appropriate Bregman divergence for this task. In addition, the optimal partitioning of the space of prior probabilities is a Bregman Voronoi diagram. Together, the optimal grouping and representation points are an \(\epsilon \) -net with respect to Bayes risk error divergence, and permit a rate-distortion type asymptotic analysis of detection performance with the number of groups. Examples of detecting signals corrupted by additive white Gaussian noise and of distinguishing exponentially-distributed signals are presented.