Abstract
The process of inverting Markov kernels relates to the important subject of Bayesian modelling and learning. In fact, Bayesian update is exactly kernel inversion. In this paper, we investigate how and when Markov kernels (aka stochastic relations, or probabilistic mappings, or simply kernels) can be inverted. We address the question both directly on the category of measurable spaces, and indirectly by interpreting kernels as Markov operators:
 For the direct option, we introduce a typed version of the category of Markov kernels and use the socalled "disintegration of measures". Here, one has to specialise to measurable spaces borne from a simple class of topological spaces e.g. Polish spaces (other choices are possible). Our method and result greatly simplify a recent development in Ref. [4].
 For the operator option, we use a cone version of the category of Markov operators (kernels seen as predicate transformers). That is to say, our linear operators are not just continuous, but are required to satisfy the stronger condition of being $\om$chaincontinuous. Prior work shows that one obtains an adjunction in the form of a pair of contravariant and inverse functors between the categories of $L_1$ and $L_\infty$cones [3]. Inversion, seen through the operator prism, is just adjunction. No topological assumption is needed.
 We show that both categories (Markov kernels and $\om$chaincontinuous Markov operators) are related by a family of contravariant functors $T_p$ for $1\leq p\leq\infty$. The $T_p$'s are Kleisli extensions of (duals of) conditional expectation functors introduced in Ref. [3].
 With this bridge in place, we can prove that both notions of inversion agree when both defined: if $f$ is a kernel, and $f\dg$ its direct inverse, then $T_\infty(f)\dg=T_1(f\dg)$.
BibTeX  Entry
@InProceedings{dahlqvist_et_al:LIPIcs:2016:6190,
author = {Fredrik Dahlqvist and Vincent Danos and Ilias Garnier and Ohad Kammar},
title = {{Bayesian Inversion by OmegaComplete Cone Duality (Invited Paper)}},
booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)},
pages = {1:11:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770170},
ISSN = {18688969},
year = {2016},
volume = {59},
editor = {Jos{\'e}e Desharnais and Radha Jagadeesan},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6190},
URN = {urn:nbn:de:0030drops61909},
doi = {10.4230/LIPIcs.CONCUR.2016.1},
annote = {Keywords: probabilistic models, bayesian learning, markov operators}
}
Keywords: 

probabilistic models, bayesian learning, markov operators 
Collection: 

27th International Conference on Concurrency Theory (CONCUR 2016) 
Issue Date: 

2016 
Date of publication: 

24.08.2016 