# A Convenient Category for Higher-Order Probability Theory

## TL;DR

This paper introduces quasi-Borel spaces as a new mathematical framework that enables higher-order probability modeling, supporting continuous distributions and higher-order functions, thus addressing limitations of traditional measure-theoretic probability.

## Contribution

The paper presents quasi-Borel spaces as a novel formalization of probability theory that is cartesian closed and supports higher-order functions and continuous distributions.

## Key findings

- Quasi-Borel spaces form a cartesian closed category for probability.
- They support continuous probability distributions.
- They enable a cleaner expression of probability constructions and generalize de Finetti's theorem.

## Abstract

Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of probability theory. Programs may use both higher-order functions and continuous distributions, or even define a probability distribution on functions. But standard probability theory does not handle higher-order functions well: the category of measurable spaces is not cartesian closed. Here we introduce quasi-Borel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and probability by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti's theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.