Cox's theorem

Implications of Cox's postulates

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

1. Certainty is represented by ${\displaystyle w(A\mid B)=1.}$
2. ${\displaystyle w^m(A|B)+w^m(\text{not }A\mid B)=1.}$
3. ${\displaystyle w(AB\mid C)=w(A\mid C)w(B\mid AC)=w(B\mid C)w(A\mid BC).}$

The postulates imply only these general properties. We may recover the usual laws of probability by setting a new function, conventionally denoted ${\displaystyle P}$ or ${\displaystyle Pr}$, equal to ${\displaystyle w^m}$. Then we obtain the laws of probability in a more familiar form:

1. Certain truth is represented by ${\displaystyle Pr(A\mid B)=1}$, and certain falsehood by ${\displaystyle Pr(A\mid B)=0.}$
2. ${\displaystyle Pr(A\mid B)+Pr(\text{not }A\mid B)=1.}$
3. ${\displaystyle Pr(AB\mid C)=Pr(A\mid C)Pr(B\mid AC)=Pr(B\mid C)Pr(A\mid BC).}$

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 (the conjunctive normal form), 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 (in which this assumption merely simplifies the calculation rather than being necessary for it) is that the probability of seeing heads for the first time after an even number of flips in a sequence of coin flips is ${\displaystyle \frac{1}{3}}$.

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 Cox (1946), which is extended with additional results and more discussion in Cox (1961). Jaynes cites Abel for the first known use of the associativity functional equation. János Aczél provides a long proof of the "associativity equation" (pages 256-267). 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 ${\displaystyle P\wedge Q}$ is a twice differentiable function ${\displaystyle f}$ of truth values of the two propositions ${\displaystyle P}$ and ${\displaystyle Q}$, i.e., ${\displaystyle T(P\wedge Q)=f(T(P),T(Q))}$, which excludes uncertainty theory's "uncertain measure" from its start, because the function ${\displaystyle f(x,y)=x\wedge y}$, used in uncertainty theory, is not differentiable with respect to ${\displaystyle x}$ and ${\displaystyle y}$. 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."

Other explicit and, especially, implicit assumptions of the theorem are challenged in papers by philosopher Mark Colyvan and mathematician and statistician Glenn Shafer.