The Compositional Structure of Bayesian Inference
Abstract
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.
Cite
@article{arxiv.2305.06112,
title = {The Compositional Structure of Bayesian Inference},
author = {Dylan Braithwaite and Jules Hedges and Toby St Clere Smithe},
journal= {arXiv preprint arXiv:2305.06112},
year = {2023}
}
Comments
Final postprint to be published in MFCS 2023. Contains material from two unpublished preprints, arxiv:2006.01631 and arXiv:2209.14728