Joint Distributions in Probabilistic Semantics

TL;DR

This paper introduces a new categorical framework for joint probability distributions in probabilistic semantics, enabling composition without disintegration and broadening applicability to various spaces.

Contribution

It proposes a category of joint finite measures with composition defined without disintegration, enhancing the semantics of probabilistic programming languages.

Findings

Defines a symmetric monoidal category of joint measures

Allows composition without disintegration restrictions

Applicable to a broader class of spaces

Abstract

Various categories have been proposed as targets for the denotational semantics of higher-order probabilistic programming languages. One such proposal involves joint probability distributions (couplings) used in Bayesian statistical models with conditioning. In previous treatments, composition of joint measures was performed by disintegrating to obtain Markov kernels, composing the kernels, then reintegrating to obtain a joint measure. Disintegrations exist only under certain restrictions on the underlying spaces. In this paper we propose a category whose morphisms are joint finite measures in which composition is defined without reference to disintegration, allowing its application to a broader class of spaces. The category is symmetric monoidal with a pleasing symmetry in which the dagger structure is a simple transpose.