# Borel Kernels and their Approximation, Categorically

## TL;DR

This paper develops a categorical framework for Borel kernels, enabling exact and approximate semantics of probabilistic programs, with applications to Bayesian inference and ProbNetKAT, and proves convergence of finite-space approximations.

## Contribution

It introduces a novel categorical structure for Borel kernels, linking kernel semantics to predicate and state transformers, and provides a method for approximating kernels on standard Borel spaces.

## Key findings

- Borel kernels form a dagger symmetric monoidal category with Bayesian inversion.
- Natural transformations relate Radon-Nikodym and Riesz theorems within this framework.
- Finite-space kernel approximations converge to Borel kernels under suitable discretizations.

## Abstract

This paper introduces a categorical framework to study the exact and approximate semantics of probabilistic programs. We construct a dagger symmetric monoidal category of Borel kernels where the dagger-structure is given by Bayesian inversion. We show functorial bridges between this category and categories of Banach lattices which formalize the move from kernel-based semantics to predicate transformer (backward) or state transformer (forward) semantics. These bridges are related by natural transformations, and we show in particular that the Radon-Nikodym and Riesz representation theorems - two pillars of probability theory - define natural transformations. With the mathematical infrastructure in place, we present a generic and endogenous approach to approximating kernels on standard Borel spaces which exploits the involutive structure of our category of kernels. The approximation can be formulated in several equivalent ways by using the functorial bridges and natural transformations described above. Finally, we show that for sensible discretization schemes, every Borel kernel can be approximated by kernels on finite spaces, and that these approximations converge for a natural choice of topology. We illustrate the theory by showing two examples of how approximation can effectively be used in practice: Bayesian inference and the Kleene star operation of ProbNetKAT.