Cox's theorem

Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. This derivation justifies the so-called "logical" interpretation of probability, as the laws of probability derived by Cox's theorem are applicable to any proposition. Logical (a.k.a. objective Bayesian) probability is a type of Bayesian probability. Other forms of Bayesianism, such as the subjective interpretation, are given other justifications.

Cox wanted his system to satisfy the following conditions:

The postulates as stated here are taken from Arnborg and Sjödin. "Common sense" includes consistency with Aristotelian logic in the sense that logically equivalent propositions shall have the same plausibility.

The postulates as originally stated by Cox were not mathematically rigorous (although better than the informal description above), e.g., as noted by Halpern. However it appears to be possible to augment them with various mathematical assumptions made either implicitly or explicitly by Cox to produce a valid proof.

Cox's notation:

Cox's postulates and functional equations are:

The laws of probability derivable from these postulates are the following. Let ${\displaystyle A\mid B}$ be the plausibility of the proposition ${\displaystyle A}$ given ${\displaystyle B}$ satisfying Cox's postulates. Then there is a function ${\displaystyle w}$ mapping plausibilities to interval [0,1] and a positive number ${\displaystyle m}$ such that

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