Derivation of the laws of probability theory
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 (also known as objective Bayesian) 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:
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 more so than the informal description above),
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:
The plausibility of a proposition formula_1 given some related information formula_2 is denoted by formula_3.
Cox's postulates and functional equations are:
In form of a functional equation
formula_12
Because of the associative nature of the conjunction in propositional logic, the consistency with logic gives a functional equation saying that the function formula_13 is an associative binary operation.
All strictly increasing associative binary operations on the real numbers are isomorphic to multiplication of numbers in a subinterval of [0, +∞], which means that there is a monotonic function formula_15 mapping plausibilities to [0, +∞] such that
formula_16
formula_21
This shall hold for any proposition formula_6, which leads to
formula_23
formula_28
This shall hold for any proposition formula_6, which, without loss of generality, leads to a solution
formula_30
Due to the requirement of monotonicity, this means that formula_15 maps plausibilities to interval [0, 1].
This postulates the existence of a function formula_32 such that
formula_33
Because "a double negative is an affirmative", consistency with logic gives a functional equation
formula_34
saying that the function formula_32 is an involution, i.e., it is its own inverse.
The above functional equations and consistency with logic imply that
formula_37
Since formula_4 is logically equivalent to formula_39, we also get
formula_40
If, in particular, formula_41, then also formula_42 and formula_43 and we get
formula_44
and
formula_45
Abbreviating formula_46 and formula_47 we get the functional equation
formula_48
Implications of Cox's postulates.
The laws of probability derivable from these postulates are the following. Let formula_49 be the plausibility of the proposition formula_1 given formula_6 satisfying Cox's postulates. Then there is a function formula_15 mapping plausibilities to interval [0,1] and a positive number formula_53 such that
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 formula_57 or formula_58, equal to formula_59. Then we obtain the laws of probability in a more familiar form:
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 formula_64.
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. 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 formula_65 is a twice differentiable function formula_32 of truth values of the two propositions formula_57 and formula_68, i.e., formula_69, which excludes uncertainty theory's "uncertain measure" from its start, because the function formula_70, used in uncertainty theory, is not differentiable with respect to formula_71 and formula_72. 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."
Notes.
 ;templatestyles src="Reflist/styles.css" / ;
References.
 ;templatestyles src="Reflist/styles.css" / ;