Bayesian Programming

Probability as an extension of logic


Cox theorem

The Cox theorem [Cox, 1946, 1961] demonstrates how the intuitive notion of plausibility can be and should be formalized by the mathematical concept of probability.

Let us note (a|c) the plausibility of a given proposition a knowing a set of knowledge c.

The following postulates explicite the plausibility notion:

  1. Plausibilities are represented by real numbers:
    (a|c) ∈ R
  2. Plausibilities are consistent:
    if there exist several correct calculi for the same plausibility, they should all lead to the same result.
  3. If some new information c′ replaces c and increases the plausibility of a, then the plausibility of the negation a ̄ should decrease:
    [(a|c′) > (a|c)] ⇒ [(a ̄|c′) < (a ̄|c)]
  4. If some new information c′ increases the plausibility of a but does not concern in any way the plausibility of b, then the plausibility of the conjunction a ∧ b should increase:
    [[(a|c′) > (a|c)] ∧ [(b|a ∧ c′) = (b|a ∧ c)]] ⇒ [(a ∧ b|c′) > (a ∧ b|c)]

Starting from these postulates, Richard T. Cox has demonstrated that plausible reasoning should follow the two rules from which all the theories can be rebuilt:

1. The normalization rule:
P (a|c) + P (a ̄|c) = 1
2. The conjunction rule:
 P (a ∧ b|c) = P (a|c) P (b|a ∧ c) = P (b|c) P (a|b ∧ c) (16.6)

Furthermore, this theorem shows that any technic for plausibility calculus that would not respect these two rules would contradict at least one of the preceding postulates. Consequently, if we accept these postulates, probability calculus is the only means to do plausible reasoning.

Chapter 2 of Jaynes’ book [2003] is completely devoted to the demonstration and discussion of Cox’s theorem and is a reference on this matter. Cox’s theorem has been partially disputed by Halpern [1999a; 1999b], himself contradicted by Arnborg and Sjödin [2000].

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