  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 work, 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.
Moreover, 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.
