Bonjour,
Les travaux de Cox, Jeffrey, Polya et d'autres ont conduit E.T Jaynes à exprimer les probabilités comme une extension de la logique booléenne : Probability Theory: The Logic of Science.
Cette approche est-elle une autre formalisation des probabilités servant, entre autre, à l'inférence statistique ou est-ce juste une interprétation épistémique des probabilités ?
THE COX THEOREM qui en est le fondement
Since the work of Laplace in the late 18th century, there have been many attempts by mathematicians to axiomitize probability theory. The most important example in the 20th century was that of A.N. Kolmogorov, who gave a very simple measure-theoretic set of axioms that modeled the view of probability introduced into quantum mechanics by Max Born in 1927. Remarkably, most physicists, in their non-quantum applications of probability, have not followed Born or Kolmogorov but R. T. Cox, who in turn base d his approach on Laplace’s original idea that probability theory is a precise mathematical formulation of plausible reasoning. These physicists argue that, while the Kolmogorov axioms are elegant and consistent, theyare much too limited in scope. In particular, the Kolmogorov axioms in their original form do not refer to conditional probabilities, whereas most physics applications of probability theory require conditional probabilities. Even though unknown by most mathematicians who work in probability theory, the Laplace-Cox approach to probability theory was actually accepted by many distinguished mathematicians prior to Kolmogorov, for examples, Augustus de Morgan, Emile Borel, Henri Poincare, and G. Polya. For a discussion of applications of Laplacian probability in the foundations and interpretation of quantum mechanics see Tipler.
We shall give in this paper a rigorous mathematical proof for Cox’s Theorem on the product rule for conditional plausibility of propositions as used in plausible reasoning, a proof that follows from precise axioms. We shall see that our axioms are mathematically simpler and more intuitive than Cox’s desiderata.
Patrick
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