From Multisets over Distributions to Distributions over Multisets

TL;DR

This paper explores the distributive law of multisets over probability distributions, revealing a rich theoretical framework that enhances understanding of combining non-determinism and probability in programming language semantics.

Contribution

It introduces the parallel multinomial law, a new distributive law of multisets over distributions, with multiple equivalent definitions and applications to sampling semantics.

Findings

The parallel multinomial law can be defined in four equivalent ways.

The law commutes with hypergeometric distributions.

Application to sampling semantics demonstrates practical relevance.

Abstract

A well-known challenge in the semantics of programming languages is how to combine non-determinism and probability. At a technical level, the problem arises from the fact that there is a no distributive law between the powerset monad and the distribution monad - as noticed some twenty years ago by Plotkin. More recently, it has become clear that there is a distributive law of the multiset monad over the distribution monad. This article elaborates the details of this distributivity and shows that there is a rich underlying theory relating multisets and probability distributions. It is shown that the new distributive law, called parallel multinomial law, can be defined in (at least) four equivalent ways. It involves putting multinomial distributions in parallel and commutes with hypergeometric distributions. Further, it is shown that this distributive law commutes with a new form of zipping for multisets. Abstractly, this can be described in terms of monoidal structure for a fixed-size multiset functor, when lifted to the Kleisli category of the distribution monad. Concretely, an application of the theory to sampling semantics is included.