# Semantics of higher-order probabilistic programs with conditioning

arXiv: https://arxiv.org/abs/1902.11189 · HTML: https://mlsanity.com/paper/1902.11189/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2019.

## Built on by (most cited first, 4 of 4)

- [Checking Trustworthiness of Probabilistic Computations in a Typed Natural Deduction System](https://mlsanity.com/paper/2206.12934/) (2022)
- [Fast and Correct Gradient-Based Optimisation for Probabilistic Programming via Smoothing](https://mlsanity.com/paper/2301.03415/) (2023)
- [Mixed Nondeterministic-Probabilistic Automata: Blending graphical probabilistic models with nondeterminism](https://mlsanity.com/paper/2201.07474/) (2022)
- [Data Petri Nets meet Probabilistic Programming (Extended version)](https://mlsanity.com/paper/2406.11883/) (2024)

## Abstract (arXiv)

We present a denotational semantics for higher-order probabilistic programs in terms of linear operators between Banach spaces. Our semantics is rooted in the classical theory of Banach spaces and their tensor products, but bears similarities with the well-known Scott semantics of higher-order programs through the use ordered Banach spaces which allow definitions in terms of fixed points. Being based on a monoidal rather than cartesian closed structure, our semantics effectively treats randomness as a resource.