Reverend Thomas Bayes (~1701-1761), pastor of the Presbyterian Church and British mathematician, studied logic and theology at the University of Edinburgh. Various works, including an introduction to differential calculus, led to his election to the *Royal Society* on November 4, 1742. It was only after Bayes's death in 1761 that his friend Richard Price found in his papers an *Essay on how to solve a problem in the doctrine of risks*. Published by the *Royal Society* in 1763, this essay applies the principle of inference that we know today as " Bayes' rule ". Without denying Bayes' inventiveness, it is now generally agreed that his rule is merely a simple application of the product rule in probability theory, already known to Bernoulli and De Moivre, and whose vast range of applications Laplace (1774) was the first to perceive.

What is it all about ? The principles of Bayesian reasoning extend the principles of classical logic with discrete truth values to continuous values of plausibility. In fact, they can be deduced from the axioms that these plausibilities must verify (Jaynes, 2003). Briefly: we assume that (1) degrees of plausibility are represented by real numbers ; (2) these values follow common-sense rules, thus following Laplace's famous formula (" probability theory is basically just common sense reduced to calculus ") ; (3) none of the available data is ignored ; (4) equivalent states of knowledge have the same degree of plausibility. The Cox-Jaynes theorem shows that these rules suffice to define, to within one monotonic function, universal mathematical rules for plausibility *p*. These are the usual rules of probability and the fundamental rule " de Bayes " :

p(A & B) = p(A|B) p(B) = p(B|A) p(A)