Cox's Theorem - Interpretation and Further Discussion

Interpretation and Further Discussion

Cox's theorem has come to be used as one of the justifications for the use of Bayesian probability theory. For example, in Jaynes it is discussed in detail in chapters 1 and 2 and is a cornerstone for the rest of the book. Probability is interpreted as a formal system of logic, the natural extension of Aristotelian logic (in which every statement is either true or false) into the realm of reasoning in the presence of uncertainty.

It has been debated to what degree the theorem excludes alternative models for reasoning about uncertainty. For example, if certain "unintuitive" mathematical assumptions were dropped then alternatives could be devised, e.g., an example provided by Halpern. However Arnborg and Sjödin suggest additional "common sense" postulates, which would allow the assumptions to be relaxed in some cases while still ruling out the Halpern example. Other approaches were devised by Hardy or Dupré and Tipler.

The original formulation of Cox's theorem is in, which is extended with additional results and more discussion in. Jaynes cites Abel for the first known use of the associativity functional equation. Aczél provides a long proof of the "associativity equation" (pages 256-267). Jaynes (p27) reproduces the shorter proof by Cox in which differentiability is assumed.