  Computation of confidence sets is central to data science and machine
learning, serving as the workhorse of A/B testing and underpinning the
operation and analysis of reinforcement learning algorithms. This paper studies
the geometry of the minimum-volume confidence sets for the multinomial
parameter. When used in place of more standard confidence sets and intervals
based on bounds and asymptotic approximation, learning algorithms can exhibit
improved sample complexity. Prior work showed the minimum-volume confidence
sets are the level-sets of a discontinuous function defined by an exact
p-value. While the confidence sets are optimal in that they have minimum
average volume, computation of membership of a single point in the set is
challenging for problems of modest size. Since the confidence sets are
level-sets of discontinuous functions, little is apparent about their geometry.
This paper studies the geometry of the minimum volume confidence sets by
enumerating and covering the continuous regions of the exact p-value function.
This addresses a fundamental question in A/B testing: given two multinomial
outcomes, how can one determine if their corresponding minimum volume
confidence sets are disjoint? We answer this question in a restricted setting.
