Ergodic Theory in Categorical Probability

We are faced with uncertainty daily, both as individuals and as societies. This has a profound impact on how we make decisions in essentially every domain of our life. Sometimes; such as when deciding whether to take an umbrella when going outside; an informal understanding is enough. In other cases; when making scientific experiments, policy decisions, or market investments; we can benefit tremendously from a formal theory of probability and statistical analysis. Traditionally, situations with uncertainty are studied using the tools of a subfield of mathematical analysis known as measure theory. This project continues the development of an alternative foundation for such situations the Synthetic Probability Theory. Measure theory starts from a precise structure of the possible events that can occur, which can be intricate and often makes the analysis inaccessible to non- experts. In contrast, the building blocks of Synthetic Probability Theory are properties that reasoning under uncertainty should satisfy, such as the ability to update knowledge given an observation or to consider a collection of independent experiments. This change of perspective enables greater generality and more intuitive explanations at the price of losing some of the nuanced distinctions. Technical details of our approach to Synthetic Probability Theory are facilitated by mathematical objects known as Markov categories, which is why we call it categorical probability. This choice has the further advantage that we can use a diagrammatic language to visualize the statements and their proofs without sacrificing any rigor. These diagrams express dependence between events and are similar to existing and widely used ways to depict statistical dependencies, which makes them particularly intuitive. In the last 5 years, the development of Synthetic Probability Theory via Markov categories has seen a rapid progress. On the one hand, many of the central theorems about uncertainty have been proven from the new perspective. Some of them have immediately obtained new applications thanks to the generality of this framework, others proofs can be now comprehended easier thanks to the intuitive diagrams. On the other hand, the language of Markov categories has been already used to model and understand causality, cognition, decision theory, machine learning, and other applications of probability theory. This project will contribute further to this effort by providing a synthetic approach to ergodic theorems. These are highly influential results that help us understand the long-term behaviour of systems that evolve over time. They tell us under which conditions the average behaviour over time matches the average behaviour across all possible initial states of the system. This idea has broad applications, from predicting weather patterns to understanding statistical physics, and even in fields like economics and data science.