Hypothesis Testing, Information Divergence and Computational Geometry

In this paper, we consider the Bayesian multiple hypothesis testing problem from the stance of computational geometry. We first recall that the probability of error of the optimal decision rule, the maximum a posteriori probability (MAP) criterion, is related to both the total variation and the Chernoff statistical distances. We then consider the exponential family manifolds, and show that the MAP rule amounts to a nearest neighbor classifier that can be implemented either by point locations in an additive Bregman Voronoi diagram or by nearest neighbor queries using various techniques of computational geometry. Finally, we show that computing the best error exponent upper bounding the probability of error, the Chernoff distance, amounts to (1) find a unique geodesic/bisector intersection point for binary hypothesis, (2) solve a closest Bregman pair problem for multiple hypothesis.