Multisets and Distributions, in Drawing and Learning

Multisets are ‘sets’ in which elements may occur multiple times. Discrete probability distributions capture states in which elements may occur with probabilities that add up to one. This paper describes how the interaction between multisets and distributions lies at the heart of some basic constructions in probability theory, especially in distributions arising from drawing from an urn with multiple balls and in learning distributions from multiple occurrences of data. Drawing multiple balls from an urn is described uniformly in terms of Kleisli iteration for a monad, covering the four standard distinctions of ordered/unordered draws, with/without replacement. In probabilistic learning the paper distinguishes two forms of likelihood, based on also on iteration, with corresponding forms of learning. Both of these forms occur in the literature, but they are not clearly distinguished, even though they lead to different outcomes.