Optimal Confidence Regions for the Multinomial Parameter

Construction of tight confidence regions and intervals is central to statistical inference and decision making. This paper develops new theory showing minimum average volume confidence regions for categorical data. More precisely, consider an empirical distribution widehatboldsymbolp generated from n iid realizations of a random variable that takes one of k possible values according to an unknown distribution boldsymbolp. This is analogous to a single draw from a multinomial distribution. A confidence region is a subset of the probability simplex that depends on widehatboldsymbolp and contains the unknown boldsymbolp with a specified confidence. This paper shows how one can construct minimum average volume confidence regions, answering a long standing question. We also show the optimality of the regions directly translates to optimal confidence intervals of linear functionals such as the mean, implying sample complexity and regret improvements for adaptive machine learning algorithms.