Categorical algebra of conditional probability
Mika Bohinen, Paolo Perrone
TL;DR
conditioning in probability is cast in a categorical framework using Markov categories and probability monads, demonstrating that conditioning arises from Beck-Chevalley via a weakly cartesian underlying functor $P$ and a weakly cartesian multiplication $μ$. The Giry monad on standard Borel spaces is shown to satisfy these BC properties, enabling a synthetic treatment of conditionals and a universal property for hypernormalizations expressed through $f:Θ→X$ and the standard measure on $PΘ$. The work links partial evaluations with second-order stochastic dominance and Blackwell order, offering a structural, diagrammatic view of statistical experiments and their information content. This approach provides a principled bridge between probability theory and categorical/descent-theoretic methods with potential implications for fibrations and higher-order probability theory.
Abstract
In the field of categorical probability, one uses concepts and techniques from category theory, such as monads and monoidal categories, to study the structures of probability and statistics. In this paper, we connect some ideas from categorical algebra, namely weakly cartesian functors and natural transformations, to the idea of conditioning in probability theory, using Markov categories and probability monads. First of all, we show that under some conditions, the monad associated to a Markov category with conditionals has a weakly cartesian functor and weakly cartesian multiplication (a condition known as Beck-Chevalley, or BC). In particular, we show that this is the case for the Giry monad on standard Borel spaces. We then connect this theory to existing results on statistical experiments. We show that for deterministic statistical experiments, the so-called standard measure construction (which can be seen as a generalization of the "hyper-normalizations" introduced by Jacobs) satisfies a universal property, allowing an equivalent definition which does not rely on the existence of conditionals.