Negative Probability
Nick Polson, Vadim Sokolov · Applied Stochastic Models in Business and Industry · 2025
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 the Bayes rule is given with negative mixing weights. The classic half‐coin distribution and Polya‐Gamma mixing are 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.