Probability and Statistics with Markov Categories

The theme of this project is the development of a new approach to the mathematical foundations of probability theory and statistics. Since the groundbreaking work of Kolmogorov in the 1930s, measure theory has been considered a suitable foundation. The starting point of this project is the idea that although Kolmogorovs axioms constitute a perfectly adequate foundation, they sometimes become difficult to use in mathematical practice when faced with complex problems. The recently developed axiom system of Markov categories has turned out to be more practical in this respect, and our goal is to develop it further. The difference to the traditional approach can be illustrated with an analogy to computer programming: although programming in machine language is possible in theory, for practical problems it is too complex to be feasible, and it is more reasonable to employ humanly comprehensible high-level programming languages. More precisely, in this project we will develop further aspects of probability theory within the framework of Markov categories. This includes in particular the law of large numbers, which is one of the central results of the classical theory and which needs to be reproduced by any alternative approach that purports to be a foundation of probability. In addition we will consider variants of the de Finetti theorem, which characterizes probability distributions with high symmetry. In existing work we have proven this theorem in the Markov categories approach, in terms of a proof that is arguably more intuitive than the classical ones in measure theory. What remains open is to similarly prove certain variants of de Finettis theorem for probability distributions with not quite as much symmetry, which have found applications to combinatorics and the theory of statistical models. In this direction of research, we also hope to be able to eventually prove new results that have not yet been obtained through the methods of measure theory.

The theory of Markov categories has seen significant development in recent years, in part thanks to this project. The main result of the project is a new abstract formulation of the law of large numbers. This classical mathematical theorem can be viewed as a self-consistency statement of probability theory, which is necessary for the interpretation of probabilities as relative frequencies. The treatment of this theorem using Markov categories is an important confirmation of their power. It will also open up new avenues for the philosophy of probability: our new axioms for "empirical sampling" provide, for the first time, the conditions under which forming relative frequencies from a sequence of outcomes is meaningful. Another important result is a new proof of the Aldous-Hoover theorem on random networks. The language of Markov categories yields a proof that is significantly more intuitive than existing ones, which rely heavily on measure theory and analysis. Thanks to this simplification, it will be possible in the future to tackle new, more complex statements about probability distributions with symmetries. Finally, the project has also contributed to the successful dissemination of the theory of Markov categories. Through talks and mini-courses at international conferences, we have been able to cater to the existing interest in the research community, leading to a growing number of researchers actively using the theory of Markov categories. This includes researchers in computer science, where Markov categories are used both in the semantic modeling of probabilistic programming languages and increasingly in machine learning.