Dependent Bayesian Lenses: Categories of Bidirectional Markov Kernels with Canonical Bayesian Inversion

TL;DR

This paper extends Bayesian Lenses to include dependent backward objects, providing a formal framework for Bayesian inversion restricted to prior-supported points within Markov categories.

Contribution

It introduces a generalized construction of Bayesian Lenses with dependent backward objects, formalizing a canonical notion of Bayesian inversion in Markov categories.

Findings

Defines support objects in Markov categories.

Establishes a section into dependent Bayesian lenses.

Provides a more canonical Bayesian inversion framework.

Abstract

We generalise an existing construction of Bayesian Lenses to admit lenses between pairs of objects where the backwards object is dependent on states on the forwards object (interpreted as probability distributions). This gives a natural setting for studying stochastic maps with Bayesian inverses restricted to the points supported by a given prior. In order to state this formally we develop a proposed definition by Fritz of a support object in a Markov category and show that these give rise to a section into the category of dependent Bayesian lenses encoding a more canonical notion of Bayesian inversion.