Multinomial and Hypergeometric Distributions in Markov Categories

TL;DR

This paper develops an abstract categorical framework for multinomial and hypergeometric distributions using Markov categories, finite colimits, and multisets, enabling a unified treatment of probabilistic draws.

Contribution

It introduces a novel categorical construction of multinomial and hypergeometric distributions on multisets within Markov categories, expanding the theoretical foundation of categorical probability.

Findings

Defined multiset functor with sums and zips in Markov categories

Established interaction of distributions with multiset operations

Provided an abstract framework for probabilistic draws using multisets

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.