This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.
Most probability, as it was understood before Bayes, ran in one direction: given a known cause, how likely is a given effect? A fair die is known to have six faces, so the probability of rolling a four is known outright. Bayes' theorem runs the calculation the other way, from an observed effect back to the probability of a candidate cause, and that reversal is what has made it, in the centuries since, a working method for medicine, spam filters, courtroom reasoning and machine learning, far outside the doctrine of chances Bayes himself was writing about in the language of eighteenth century probability. In its plainest form the theorem says the probability of a hypothesis, given some observed evidence, depends on three things: how probable the evidence would be if the hypothesis were true, how probable the hypothesis seemed before the evidence arrived, and how probable the evidence was overall, however it came about. A medical test that is genuinely reliable, correctly positive nineteen times out of twenty, can still make a positive result more likely to be a false alarm than a true finding, if the condition it tests for is rare enough in the population being tested. That is not a paradox in the theorem, it is exactly what the theorem predicts once the low starting probability of the rare condition is folded in, and it is the single most common way Bayes' theorem gets invoked to correct an intuition that ignores how rare something was to begin with. The Bayesian approach to statistics generally, the practice of starting from a stated prior belief and updating it in light of new data using this same reasoning, is named for the theorem and remains one of two major competing philosophies of what probability itself means, standing against the frequentist tradition that treats probability only as a long-run frequency.