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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 propositions, logical probability is a variety of Bayesian probability. Other forms of Bayesianism, such as the subjective interpretation, are given other justifications.
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Cox's assumptions
Cox wanted his system to satisfy the following desiderata
- Divisibility and comparability - The plausibility of a statement is a real number and is dependent on information we have related to the statement.
- Common sense - Plausibilities should vary sensibly with the assessment of plausibilities in the model.
- Consistency - If the plausibility of a statement can be derived in two ways, the two results must be equal.
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
-
- f(f(x)) = x,
- saying that the function f that maps the probability of a proposition to the probability of the proposition's negation is an involution, i.e., it is its own inverse.
- 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 multiplication of probabilities 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
- is an associative binary operation. All strictly increasing associative binary operations on the real numbers are isomorphic to multiplication of numbers in the interval 0, 1. This function therefore may be taken to be multiplication.
- 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
- where f is as above.
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.
- Certainty is represented by w(A|B) = 1.
- wm(A|B) + wm(AC|B) = 1
- w(A, B|C) = w(A|C) w(B|A, C) = w(B|C) w(A|B, C)
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:
- Certain truth is represented by Pr(A|B) = 1, and certain falsehood by Pr(A|B) = 0.
- Pr(A|B) + Pr(AC|B) = 1
- Pr(A, B|C) = Pr(A|C) Pr(B|A, C) = Pr(B|C) Pr(A|B, C)
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 the proof of certain theorems, however, it is not clear what difference countable additivity makes in practice.
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 (2003) 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 (1999a). However Arnborg and Sjödin (1999, 2000a, 2000b) suggest additional "common sense" postulates, which would allow the assumptions to be relaxed in some cases while still ruling out the Halpern example.
The original formulation of Cox's theorem is in Cox (1946), which is extended with additional results and more discussion in Cox (1961). Jaynes (2003) cites Abel (1826) as first known instance of the associativity functional equation which is used in the proof of the theorem. Aczél (1966) refers to the "associativity equation" and lists 98 references to works that discuss it or use it, and gives a proof that doesn't require differentiability (pages 256-267).
References and external links
- Niels Henrik Abel "Untersuchung der Functionen zweier unabhängig veränderlichen Gröszen x und y, wie f(x, y), welche die Eigenschaft haben, dasz fz, f(x,y) eine symmetrische Function von z, x und y ist.", Jour. Reine u. angew. Math. (Crelle's Jour.), 1, 11-15, (1826).
- R. T. Cox, "Probability, Frequency, and Reasonable Expectation," Am. Jour. Phys., 14, 1-13, (1946).
- R. T. Cox, The Algebra of Probable Inference, Johns Hopkins University Press, Baltimore, MD, (1961).
- Janos Aczél, Lectures on Functional Equations and their Applications, Academic Press, New York, (1966).
- Terrence L. Fine, Theories of Probability; An examination of foundations, Academic Press, New York, (1973).
- Edwin Thompson Jaynes, Probability Theory: The Logic of Science, Cambridge University Press (2003). -- preprint version (1996) at http://omega.albany.edu:8008/JaynesBook.html; Chapters 1 to 3 of published version at http://bayes.wustl.edu/etj/prob/book.pdf
- Joseph Y. Halpern, "A counterexample to theorems of Cox and Fine," Journal of AI research, 10, 67-85 (1999) -- http://www.cs.washington.edu/research/jair/abstracts/halpern99a.html
- Joseph Y. Halpern, "Technical Addendum, Cox's theorem Revisited," Journal of AI research, 11, 429-435 (1999) -- http://www.cs.washington.edu/research/jair/abstracts/halpern99b.html
- Stefan Arnborg and Gunnar Sjödin, On the foundations of Bayesianism, Preprint: Nada, KTH (1999) -- ftp://ftp.nada.kth.se/pub/documents/Theory/Stefan-Arnborg/06arnborg.ps -- ftp://ftp.nada.kth.se/pub/documents/Theory/Stefan-Arnborg/06arnborg.pdf
- Stefan Arnborg and Gunnar Sjödin, A note on the foundations of Bayesianism, Preprint: Nada, KTH (2000a) -- ftp://ftp.nada.kth.se/pub/documents/Theory/Stefan-Arnborg/fobshle.ps -- ftp://ftp.nada.kth.se/pub/documents/Theory/Stefan-Arnborg/fobshle.pdf
- Stefan Arnborg and Gunnar Sjödin, "Bayes rules in finite models," in European Conference on Artificial Intelligence, Berlin, (2000b) -- ftp://ftp.nada.kth.se/pub/documents/Theory/Stefan-Arnborg/fobc1.ps -- ftp://ftp.nada.kth.se/pub/documents/Theory/Stefan-Arnborg/fobc1.pdf
- Michael Hardy, in Advances in Applied Mathematics August 2002, pages 243-292 (or preprint) "I assert there that I think Cox's assumptions are too strong, although I don't really say why. I do say what I would replace them with." (The quote is from a Wikipedia discussion page, not from the article.)