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How Kirkwood and Probability Distributions Differ: A Coxian Perspective

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.