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Negative probabilities arise primarily in quantum theory and computing. Bartlett provides a definition based on characteristic functions and extraordinary random variables. As Bartlett observes, negative probabilities must always be combined with positive probabilities to yield a valid probability distribution before any physical interpretation is admissible. Negative probabilities arise as mixing distributions of unobserved latent variables in Bayesian modeling. Our goal is to provide a link with dual densities and the class of scale mixtures of normal distributions. We provide an analysis of the classic half coin distribution and Feynman's negative probability examples. A number of examples of dual densities with negative mixing measures including the linnik distribution, Wigner distribution and the stable distribution are provided. Finally, we conclude with directions for future research.
Polson et al. (Sun,) studied this question.