Bayes' Formula

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Abstract
The Reverend Thomas Bayes led a quiet, celibate life as a Presbyterian minister in Tunbridge Wells, Kent, England, in the middle years of the 1700s. As he strolled the gentle hills of southern England, pondering clerical matters, he also must have reflected upon the secular topic of probability, "chances" to use his word. Some of his recorded ideas about probability were sufficiently compelling that his friend, the Reverend Richard Price, presented them to the Royal Society in 1763, two years after Bayes' death.1 The "essay" that Price presented on behalf of his late friend contained Bayes' ideas about the effect of new information on previously held beliefs about "chances." One might hold prior beliefs about chance events, say coin flips, before witnessing any such events. These are called "a priori beliefs" or, with coin flips, "probabilities" or "odds." After some flips are observed, the belief about the likelihood of