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We develop Markov categories as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs. This means that we treat the following concepts in purely abstract categorical terms: conditioning…
This paper considers the difficulty in the set-system approach to generalizing graph theory. These difficulties arise categorically as the category of set-system hypergraphs is shown not to be cartesian closed and lacks enough projective…
The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory,…
We show that various functionals related to the supremum of a real function defined on an arbitrary set or a measure space are Hadamard directionally differentiable. We specifically consider the supremum norm, the supremum, the infimum, and…
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration…
We exhibit a functor from the category OUS of order unit spaces and positive, unit-preserving mappings into the category $\Prob$ of probabilistic models (test spaces with designated state spaces) and morphisms thereof. Restricted to any…
Forecasts of multivariate probability distributions are required for a variety of applications. Scoring rules enable the evaluation of forecast accuracy, and comparison between forecasting methods. We propose a theoretical framework for…
We determine an explicit Gr\"obner basis, consisting of linear forms and determinantal quadrics, for the prime ideal of Raftery's mixture transition distribution model for Markov chains. When the states are binary, the corresponding…
Hypergeometric functions of complex matrices were introduced by James in multivariate statistics. These special functions play many roles in random matrix theory. The main goal of this paper is to suggest a new use for them as holomorphic…
Hypersubstitutions are mappings which map operation symbols to terms. Terms can be visualized by trees. Hypersubstitutions can be extended to mappings defined on sets of trees. The nodes of the trees, describing terms, are labelled by…
We explore the class of exchangeable Bernoulli distributions building on their geometrical structure. Exchangeable Bernoulli probability mass functions are points in a convex polytope and we have found analytical expressions for their…
A new family of tree-structured Markov random fields for a vector of discrete counting random variables is introduced. According to the characteristics of the family, the marginal distributions of the Markov random fields are all Poisson…
Functions with uniform sublevel sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used in multicriteria optimization, decision theory, mathematical…
We study a category of probability spaces and measure-preserving Markov kernels up to almost sure equality. This category contains, among its isomorphisms, mod-zero isomorphisms of probability spaces. It also gives an isomorphism between…
The main result of this paper utilizes the representation graph of a group $G$, $R(V,G)$, and gives a general construction of a diagrammatic category $\mathbf{Dgrams}_{R(V,G)}$. The proof of the main theorem shows that, given explicit…
Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods…
Mal'tsev categories turned out to be a central concept in categorical algebra. On one hand, the simplicity and the beauty of the notion is revealed through a lot of characterizations of different flavour. Depending on the context, one can…
We initiate the systematic study of modular representations of symmetric groups that arise via the braiding in (symmetric) tensor categories over fields of positive characteristic. We determine what representations appear for certain…
A key problem in the application of first-order probabilistic methods is the enormous size of graphical models they imply. The size results from the possible worlds that can be generated by a domain of objects and relations. One of the…
We study the connections existing between max-infinitely divisible distributions and Poisson processes from the point of view of functional analysis. More precisely, we derive functional identities for the former by using well-known results…