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 probability is a type of Bayesian probability. Other forms of Bayesianism, such as the subjective interpretation, are given other justifications.

Cox's assumptions

Cox wanted his system to satisfy the following conditions:
  1. Divisibility and comparability - The plausibility of a proposition is a real number and is dependent on information we have related to the proposition.
  2. Common sense - Plausibilities should vary sensibly with the assessment of plausibilities in the model.
  3. Consistency - If the plausibility of a proposition can be derived in many ways, all the results must be equal.
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,
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 plausibility of the conjunction of two propositions,, given some related information, is determined by the plausibility of given and that of given.
  • Additionally, Cox postulates the function to be monotonic.
  • In case given is certain, we have and due to the requirement of consistency. The general equation then leads to
  • In case given is impossible, we have and due to the requirement of consistency. The general equation then leads to
  • The plausibility of a proposition determines the plausibility of the proposition's negation.
  • Furthermore, Cox postulates the function to be monotonic.

Implications of Cox's postulates

The laws of probability derivable from these postulates are the following. Let be the plausibility of the proposition given satisfying Cox's postulates. Then there is a function mapping plausibilities to interval and a positive number such that
  1. Certainty is represented by
It is important to note that the postulates imply only these general properties. We may recover the usual laws of probability by setting a new function, conventionally denoted or, equal to. Then we obtain the laws of probability in a more familiar form:
  1. Certain truth is represented by, and certain falsehood by
Rule 2 is a rule for negation, and rule 3 is a rule for conjunction. Given that any proposition containing conjunction, disjunction, and negation can be equivalently rephrased using conjunction and negation alone, we can now handle any compound proposition.
The laws thus derived yield finite additivity of probability, but not countable additivity. The measure-theoretic formulation of Kolmogorov assumes that a probability measure is countably additive. This slightly stronger condition is necessary for certain results. An elementary example is that the probability of seeing heads for the first time after an even number of flips in a sequence of coin flips is.

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 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. János Aczél provides a long proof of the "associativity equation". Jaynes reproduces the shorter proof by Cox in which differentiability is assumed. A guide to Cox's theorem by Van Horn aims at comprehensively introducing the reader to all these references.
Baoding Liu, the founder of uncertainty theory, criticizes Cox's theorem for presuming that the truth value of conjunction is a twice differentiable function of truth values of the two propositions and, i.e.,, which excludes uncertainty theory's "uncertain measure" from its start, because the function, used in uncertainty theory, is not differentiable with respect to and. According to Liu, "there does not exist any evidence that the truth value of conjunction is completely determined by the truth values of individual propositions, let alone a twice differentiable function."