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Joint Distributions in Probabilistic Semantics

Dexter Kozen, Alexandra Silva, Erik Voogd

TL;DR

This work introduces a category, $ ext{JDist}$, whose morphisms are joint distributions with fixed marginals, and defines composition without relying on disintegration, thereby broadening the semantic foundation for higher-order probabilistic languages. It proves a discrete, disintegration-free composition formula and extends these ideas to a general setting via approximants, culminating in a dagger-symmetric monoidal category in which the dagger is simply transpose. The paper shows a faithful embedding of the Markov-kernel category $ ext{Krn}$ into $ ext{JDist}$ and clarifies conditions under which this embedding is full, notably for standard Borel spaces. This framework enables semantics and approximation schemes for probabilistic programs on more general spaces, with potential for point-free formulations and robust handling of null sets. Overall, the approach provides a flexible, mathematically principled alternative to disintegration-based semantics with practical implications for probabilistic programming and Bayesian inference.

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.

Joint Distributions in Probabilistic Semantics

TL;DR

This work introduces a category, , whose morphisms are joint distributions with fixed marginals, and defines composition without relying on disintegration, thereby broadening the semantic foundation for higher-order probabilistic languages. It proves a discrete, disintegration-free composition formula and extends these ideas to a general setting via approximants, culminating in a dagger-symmetric monoidal category in which the dagger is simply transpose. The paper shows a faithful embedding of the Markov-kernel category into and clarifies conditions under which this embedding is full, notably for standard Borel spaces. This framework enables semantics and approximation schemes for probabilistic programs on more general spaces, with potential for point-free formulations and robust handling of null sets. Overall, the approach provides a flexible, mathematically principled alternative to disintegration-based semantics with practical implications for probabilistic programming and Bayesian inference.

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.
Paper Structure (11 sections, 6 theorems, 53 equations, 1 figure)

This paper contains 11 sections, 6 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Illustrative Example
  3. Discrete Approach
  4. Preliminaries
  5. Discrete Markov Kernels to Joint Distributions
  6. $\Krn$ and $\JDist$
  7. Preliminaries
  8. From $\Krn$ to $\JDist$
  9. Approximants
  10. Composition in $\JDist$
  11. Conclusion

Key Result

lemma 1

Let $(X,\AA)$ and $(Y,\BB)$ be measurable spaces, $\BB$ countably generated. Let $P,Q:X\to Y$ be Markov kernels and $\mu$ a probability measure on $X$. The following are equivalent:

Theorems & Definitions (12)

  • lemma 1
  • proof
  • lemma 2
  • proof
  • theorem 1: Lebesgue-
  • proof
  • theorem 2
  • proof
  • corollary 1
  • proof
  • ...and 2 more