A Higher-Order Language for Markov Kernels and Linear Operators

TL;DR

This paper introduces a new higher-order language that unifies Markov kernel and linear operator semantics for probabilistic programming, enabling combined reasoning about probabilistic programs.

Contribution

It defines a two-level calculus with categorical semantics that integrates both semantics styles, offering a novel resource interpretation of linear logic focused on sampling.

Findings

Unified framework for Markov kernels and linear operators

Enables programming with both semantics styles

Provides categorical semantics for the combined language

Abstract

Much work has been done to give semantics to probabilistic programming languages. In recent years, most of the semantics used to reason about probabilistic programs fall in two categories: semantics based on Markov kernels and semantics based on linear operators. Both styles of semantics have found numerous applications in reasoning about probabilistic programs, but they each have their strengths and weaknesses. Though it is believed that there is a connection between them there are no languages that can handle both styles of programming. In this work we address these questions by defining a two-level calculus and its categorical semantics which makes it possible to program with both kinds of semantics. From the logical side of things we see this language as an alternative resource interpretation of linear logic, where the resource being kept track of is sampling instead of variable use.