# Semantics of higher-order probabilistic programs with conditioning

## TL;DR

This paper develops a denotational semantics for higher-order probabilistic programs using linear operators on Banach spaces, emphasizing the treatment of randomness as a resource within a monoidal framework.

## Contribution

It introduces a novel semantics based on Banach space theory that captures higher-order probabilistic computations with conditioning.

## Key findings

- Semantics based on linear operators between Banach spaces.
- Framework allows fixed point definitions for higher-order programs.
- Treats randomness as a resource within a monoidal structure.

## Abstract

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.