# The Compositional Structure of Bayesian Inference

arXiv: https://arxiv.org/abs/2305.06112 · HTML: https://mlsanity.com/paper/2305.06112/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 21 references that are in the map.
- First posted on arXiv 10 May 2023.

## Builds on (most cited first, 10 of 10)

- [An Introduction to Probabilistic Programming](https://mlsanity.com/paper/1809.10756/) (2018)
- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Predictive Coding: Towards a Future of Deep Learning beyond Backpropagation?](https://mlsanity.com/paper/2202.09467/) (2022)
- [Categorical Foundations of Gradient-Based Learning](https://mlsanity.com/paper/2103.01931/) (2021)
- [Disintegration and Bayesian Inversion via String Diagrams](https://mlsanity.com/paper/1709.00322/) (2017)
- [On the relationship between predictive coding and backpropagation](https://mlsanity.com/paper/2106.13082/) (2021)
- [Compositional game theory](https://mlsanity.com/paper/1603.04641/) (2016)
- [Value Iteration is Optic Composition](https://mlsanity.com/paper/2206.04547/) (2022)
- [Mathematical Foundations for a Compositional Account of the Bayesian Brain](https://mlsanity.com/paper/2212.12538/) (2022)
- [Dependent Bayesian Lenses: Categories of Bidirectional Markov Kernels with Canonical Bayesian Inversion](https://mlsanity.com/paper/2209.14728/) (2022)

## Built on by (most cited first, 3 of 3)

- [Position: Categorical Deep Learning is an Algebraic Theory of All Architectures](https://mlsanity.com/paper/2402.15332/) (2024)
- [Reinforcement Learning in Categorical Cybernetics](https://mlsanity.com/paper/2404.02688/) (2024)
- [AutoBayes: A Compositional Framework for Generalized Variational Inference](https://mlsanity.com/paper/2503.18608/) (2025)

## Abstract (arXiv)

Bayes' rule tells us how to invert a causal process in order to update our beliefs in light of new evidence. If the process is believed to have a complex compositional structure, we may observe that the inversion of the whole can be computed piecewise in terms of the component processes. We study the structure of this compositional rule, noting that it relates to the lens pattern in functional programming. Working in a suitably general axiomatic presentation of a category of Markov kernels, we see how we can think of Bayesian inversion as a particular instance of a state-dependent morphism in a fibred category. We discuss the compositional nature of this, formulated as a functor on the underlying category and explore how this can used for a more type-driven approach to statistical inference.