Introduction to categorical probability

In recent decades, categorical methods have gained increasing attention in theoretical computer science, where programs are often understood in terms of composable building blocks. This approach contrasts with standard probability theory, where compositional reasoning is expressed indirectly through operations such as integration, summation, and product. Categorical probability, realized for instance via Markov categories, provides a bridge between the two perspectives, by formulating probabilistic reasoning in explicitly compositional terms. This field offers useful conceptual insights in probabilistic programming and has begun to influence foundational views on AI.

Perhaps surprisingly, this language is not merely a reformulation of standard probability theory: it enables rigorous discussions of several foundational results, such as the d-separation criterion for Bayesian networks and the de Finetti theorem. Moreover, different possibilistic and probabilistic perspectives can be considered as examples of the same framework, given by Markov categories, allowing results to be proved for all at once; this is the case of the d-separation criterion.

In this seminar, I will provide an overview of the field, beginning with standard probabilistic definitions, such as determinism, almost-sure equality, and Bayesian inversion. I will then discuss how one can reason from this abstract perspective. Time permitting, more advanced material will be presented, including the Kolmogorov extension theorem.