Title:Negative Probability

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.