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 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 (1999). "Common sense" includes consistency with Aristotelian logic when statements are completely plausible or implausible.
The postulates as originally stated by Cox were not mathematically rigorous (although better than the informal description above), e.g., as noted by Halpern (1999a, 1999b). 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 axioms and functional equations are:
- The plausibility of a proposition determines the plausibility of the proposition's negation; either decreases as the other increases. Because "a double negative is an affirmative", this becomes a functional equation
- The plausibility of the conjunction [A & B] of two propositions A, B, depends only on the plausibility of B and that of A given that B is true. (From this Cox eventually infers that conjunction of plausibilities is associative, and then that it may as well be ordinary multiplication of real numbers.) Because of the associative nature of the "and" operation in propositional logic, this becomes a functional equation saying that the function g such that
- Suppose [A & B] is equivalent to [C & D]. If we acquire new information A and then acquire further new information B, and update all probabilities each time, the updated probabilities will be the same as if we had first acquired new information C and then acquired further new information D. In view of the fact that multiplication of probabilities can be taken to be ordinary multiplication of real numbers, this becomes a functional equation
Cox's theorem implies that any plausibility model that meets the postulates is equivalent to the subjective probability model, i.e., can be converted to the probability model by rescaling.
Implications of Cox's postulates
The laws of probability derivable from these postulates are the following (Jaynes, 2003). Here w(A|B) is the "plausibility" of the proposition A given B, and m is some positive number.
It is important to note that the postulates imply only these general properties. These are equivalent to the usual laws of probability assuming some conventions, namely that the scale of measurement is from zero to one, and the plausibility function, conventionally denoted P or Pr, is equal to wm. (We could have equivalently chosen to measure probabilities from one to infinity, with infinity representing certain falsehood.) With these conventions, we obtain the laws of probability in a more familiar form:
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