Generalized probabilities in statistical theories

TL;DR

This review explores various formal frameworks for generalized probabilities in statistical theories, including classical, quantum, and non-commutative models, highlighting their mathematical foundations and potential generalizations.

Contribution

It systematically compares different approaches to generalized probabilities, emphasizing their mathematical structures and potential extensions beyond classical frameworks.

Findings

Classical and quantum probabilities are special cases within broader formal frameworks.

Non-commutative models extend traditional probability theories.

Convex set approaches offer a unified perspective on generalized probabilities.

Abstract

In this review article we present different formal frameworks for the description of generalized probabilities in statistical theories. We discuss the particular cases of probabilities appearing in classical and quantum mechanics, possible generalizations of the approaches of A. N. Kolmogorov and R. T. Cox to non-commutative models, and the approach to generalized probabilities based on convex sets.