Interpretations of Negative Probabilities

In this paper, we give a frequency interpretation of negative probability, as well as of extended probability, demonstrating that to a great extent, these new types of probabilities, behave as conventional probabilities. Extended probability comprises both conventional probability and negative probability. The frequency interpretation of negative probabilities gives supportive evidence to the axiomatic system built in (Burgin, 2009; arXiv:0912.4767) for extended probability as it is demonstrated in this paper that frequency probabilities satisfy all axioms of extended probability.

💡 Research Summary

The paper “Interpretations of Negative Probabilities” attempts to place the concept of negative and extended probabilities on a firm frequentist footing and to show that this frequentist construction satisfies the axioms of extended probability introduced by Burgin (2009). After a brief historical survey of how negative probabilities first appeared in quantum mechanics (Dirac, Heisenberg, Wigner) and later in various applied fields such as machine learning and finance, the author reviews the three main families of probability interpretations—objective (frequency), subjective (belief), and combined (evidence‑supported)—and argues that the objective, frequency‑based view is the most natural for physics‑originated concepts like negative probability.

The core of the paper is a formal definition of an “extended probability space.” The sample space Ω is split into two disjoint irreducible parts, Ω⁺ (positive elementary events) and Ω⁻ (their antievents). A collection F of subsets of Ω is required to be closed under a special binary operation called “union with annihilation”:

 X + Y ≡ (X ∪ Y) \