# The Geometry of Bayesian Programming

## TL;DR

This paper introduces a geometric interaction model for a typed lambda-calculus designed for higher-order Bayesian programming, incorporating sampling and soft conditioning, based on measurable spaces.

## Contribution

It provides a novel geometric interaction model for Bayesian programming languages, connecting category theory with probabilistic semantics.

## Key findings

- Model is adequate for distribution-based semantics
- Model is adequate for sampling-based semantics
- Framework supports higher-order Bayesian programming

## Abstract

We give a geometry of interaction model for a typed lambda-calculus endowed with operators for sampling from a continuous uniform distribution and soft conditioning, namely a paradigmatic calculus for higher-order Bayesian programming. The model is based on the category of measurable spaces and partial measurable functions, and is proved adequate with respect to both a distribution-based and a sampling based operational semantics.