How Kirkwood and Probability Distributions Differ: A Coxian Perspective

TL;DR

This paper explores the differences between Kirkwood and probability distributions from a Coxian perspective, proposing a rule for complex probabilities and showing that certain quasiprobability distributions cannot rank plausibility.

Contribution

It introduces a rule for complex probabilities consistent with Coxian and Bayesian principles and demonstrates that Kirkwood and Hofmann distributions violate this rule.

Findings

Kirkwood and Hofmann distributions do not obey the plausibility-ranking rule.

Complex probabilities can be consistent with Coxian interpretation under specific rules.

Certain quasiprobability distributions cannot serve as plausibility measures.

Abstract

Kolmogorov's first axiom of probability is probability takes values between 0 and 1; however, in Cox's derivation of probability having a maximum value of unity is arbitrary since he derives probability as a tool to rank degrees of plausibility. Probability can then be used to make inferences in instances of incomplete information, which is the foundation of Baysian probability theory. This article formulates a rule, which if obeyed, allows probability to take complex values and still be consistent with the interpretation of probability theory as being a tool to rank plausibility. It is then shown that Kirkwood distributions and the conditional complex probability distributions proposed by Hofmann do not obey this rule and therefore cannot rank plausibility. Not only do these quasiprobability distributions relax Kolmogorov's first axiom of probability, they also are void of the defining property of a probability distribution from a Coxian and Baysian perspective - they lack the ability to rank plausibility.