Negative Probability

TL;DR

This paper explores the concept of negative probabilities across physics and Bayesian modeling, linking different perspectives and discussing their properties, examples, and implications for quantum mechanics and statistical inference.

Contribution

It provides a unified framework connecting negative probabilities in physics and Bayesian models, including definitions, properties, and examples of dual distributions with negative mixing measures.

Findings

Negative probabilities can be modeled with extraordinary random variables.

Examples include the Linnik and Wigner distributions.

A version of Bayes rule with negative weights is discussed.

Abstract

Negative probabilities arise primarily in physics, statistical quantum mechanics and quantum computing. Negative probabilities arise as mixing distributions of unobserved latent variables in Bayesian modeling. Our goal is to provide a link between these two viewpoints. Bartlett provides a definition of negative probabilities based on extraordinary random variables and properties of their characteristic function. A version of Bayes rule is given with negative mixing weights. The classic half coin distribution and Polya-Gamma mixing is discussed. Heisenberg's principle of uncertainty and the duality of scale mixtures of Normals is also discussed. A number of examples of dual densities with negative mixing measures are provided including the Linnik and Wigner distributions. Finally, we conclude with directions for future research.