Multinomial and Hypergeometric Distributions in Markov Categories

Abstract

Markov categories, having tensors with copying and discarding, provide a setting for categorical probability. This paper uses finite colimits and what we call uniform states in such Markov categories to define a (fixed size) multiset functor, with basic operations for sums and zips of multisets, and a graded monad structure. Multisets can be used to represent both urns filled with coloured balls and also draws of multiple balls from such urns. The main contribution of this paper is the abstract definition of multinomial and hypergeometric distributions on multisets, as draws. It is shown that these operations interact appropriately with various operations on multisets.1. Probabilistic programming languages that incorporate updating (conditioning) and/or higher order features, see e.g. [10,11,12,32,34,13].2. The compositional approach to Bayesian networks [8,16] and to Bayesian reasoning [9,27,25].3. The use of diagrammatic methods in (quantum) foundations and probability, see [7] for an overview.4. Study of 'probability monads', e.g. in [30,23]. Axiomatisation of disintegration as key probabilistic technique, see e.g. [17,3,19,18], and also [5].6. Exploration of categorical structures, such as compact closed categories [1,33] or effectuses [21,4].These topics cover both ordinary (classical) probability as well as quantum probability.An issue that we are particularly interested in is the interplay between multisets (a.k.a. bags) and (probability) distributions (see e.g. [26]). Multisets play a fundamental role in probability theory, for instance as representations of urns with coloured balls, and also of draws from such urns. More generally, in learning, collections of data items, possibly occurring multiple times, are properly represented as multisets. Multinomial and hypergeometric distributions assign probabilities to draws from an urn,

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