Domains and Stochastic Processes

TL;DR

This paper develops a domain-theoretic framework for probability and stochastic processes, replacing classical concepts with domain-theoretic analogs to enhance understanding of probabilistic programming and Bayesian reasoning.

Contribution

It introduces a novel domain-theoretic approach to probability theory, recasting key stochastic process results like Skorohod's Theorem in this new framework.

Findings

Reformulation of Skorohod's Theorem in domain-theoretic terms

Potential for improved probabilistic models in computation

Foundation for probabilistic programming languages

Abstract

Domain theory has a long history of applications in theoretical computer science and mathematics. In this article, we explore the relation of domain theory to probability theory and stochastic processes. The goal is to establish a theory in which Polish spaces are replaced by domains, and measurable maps are replaced by Scott-continuous functions. We illustrate the approach by recasting one of the fundamental results of stochastic process theory -- Skorohod's Representation Theorem -- in domain-theoretic terms. We anticipate the domain-theoretic version of results like Skorohod's Theorem will improve our understanding of probabilistic choice in computational models, and help devise models of probabilistic programming, with its focus on programming languages that support sampling from distributions where the results are applied to Bayesian reasoning.