@article{TEXTUAL,
      recid = {14489},
      author = {Polson, Nick and Sokolov, Vadim},
      title = {Negative Probability},
      journal = {Applied Stochastic Models in Business and Industry},
      address = {2025-01-29},
      number = {TEXTUAL},
      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.},
      url = {http://knowledge.uchicago.edu/record/14489},
}