Related papers: Random Variables, Conditional Independence and Cat…
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…
We develope the framework of transitional conditional independence. For this we introduce transition probability spaces and transitional random variables. These constructions will generalize, strengthen and unify previous notions of…
Markov categories are a recent categorical approach to the mathematical foundations of probability and statistics. Here, this approach is advanced by stating and proving equivalent conditions for second-order stochastic dominance, a widely…
Conditional independence and Markov properties are powerful tools allowing expression of multidimensional probability distributions by means of low-dimensional ones. As multidimensional possibilistic models have been studied for several…
Markov kernels play a decisive role in probability and mathematical statistics theories, and are an extension of the concepts of sigma-field and statistic. Concepts such as independence, sufficiency, completeness, ancillarity or conditional…
We present a novel, yet rather simple construction within the traditional framework of Scott domains to provide semantics to probabilistic programming, thus obtaining a solution to a long-standing open problem in this area. Unlike current…
We introduce the categories of quasi-measurable spaces, which are slight generalizations of the category of quasi-Borel spaces, where we now allow for general sample spaces and less restrictive random variables, spaces and maps. We show…
The data of a physical experiment can be represented as a presheaf of probability distributions. A striking feature of quantum theory is that those probability distributions obtained in quantum mechanical experiments do not always admit a…
We consider deep multivariate models for heterogeneous collections of random variables. In the context of computer vision, such collections may e.g. consist of images, segmentations, image attributes, and latent variables. When developing…
A Markov network characterizes the conditional independence structure, or Markov property, among a set of random variables. Existing work focuses on specific families of distributions (e.g., exponential families) and/or certain structures…
Independence and conditional independence are fundamental concepts for reasoning about groups of random variables in probabilistic programs. Verification methods for independence are still nascent, and existing methods cannot handle…
We use Markov categories to generalize the basic theory of Markov chains and hidden Markov models to an abstract setting. This comprises characterizations of hidden Markov models in terms of conditional independences and algorithms for…
We study notions of robustness of Markov kernels and probability distribution of a system that is described by $n$ input random variables and one output random variable. Markov kernels can be expanded in a series of potentials that allow to…
Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to Markov categories…
Markov networks are popular models for discrete multivariate systems where the dependence structure of the variables is specified by an undirected graph. To allow for more expressive dependence structures, several generalizations of Markov…
Conditional independence and graphical models are well studied for probability distributions on product spaces. We propose a new notion of conditional independence for any measure $\Lambda$ on the punctured Euclidean space $\mathbb…
The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion…
This paper examines the foundational concept of random variables in probability theory and statistical inference, demonstrating that their mathematical definition requires no reference to randomization or hypothetical repeated sampling. We…
Dependency networks (Heckerman et al., 2000) provide a flexible framework for modeling complex systems with many variables by combining independently learned local conditional distributions through pseudo-Gibbs sampling. Despite their…
We study finite probability theory through a category of finite probability schemes and probability-preserving maps, called \emph{bundles}. A bundle simultaneously records a quotient of a sample space, an algebra of random variables, and…