Bayesian Posterior Probability Estimator

Bayes theorem converts a prior belief into a posterior belief by weighting it with how well each hypothesis predicted the data: P(H|D) equals P(D|H) times P(H), divided by the total probability of the data. Here the two competing hypotheses are point values for a Bernoulli success probability, so the likelihood of k successes in n trials is the binomial term C(n,k) times p to the k times (1 - p) to the (n - k). The binomial coefficient is identical under both hypotheses and cancels in the Bayes factor, which is why the ratio depends only on the two success probabilities and the observed counts. Harold Jeffreys proposed the interpretive scale still in use: a Bayes factor between 1 and 3 is barely worth mentioning, 3 to 10 substantial, 10 to 30 strong, 30 to 100 very strong, and above 100 decisive. The most instructive property of the calculation is how heavily the prior dominates when base rates are extreme. A screening test with 95 percent sensitivity and a 10 percent false positive rate, applied to a condition with 1 percent prevalence, yields a posterior of only about 8.8 percent after a single positive result - the base rate neglect that Kahneman and Tversky documented experimentally. The limitations are that this model compares two specific parameter values rather than integrating over a continuous prior, and that with small n the answer is driven mostly by the prior, so a sensitivity analysis across a range of plausible priors is more informative than any single posterior figure.

The rule is named for Thomas Bayes, a Presbyterian minister and mathematician whose essay on the doctrine of chances was found among his papers after his death in 1761 and read to the Royal Society by Richard Price in 1763. Pierre-Simon Laplace arrived at the same result independently in 1774 and did far more to develop it, applying inverse probability to astronomy, demography and jurisprudence in his Theorie analytique des probabilites of 1812. The method then fell out of favour. Ronald Fisher and Jerzy Neyman rejected inverse probability during the 1920s and 1930s on the grounds that priors were arbitrary, and frequentist significance testing dominated academic statistics for decades afterwards. It survived in applied work: Alan Turing and I J Good used sequential weight-of-evidence calculations against Enigma traffic at Bletchley Park during the Second World War, measuring evidence in units Turing called bans and decibans, and Good set out the framework publicly in Probability and the Weighing of Evidence in 1950. Harold Jeffreys had already made the theoretical case in Theory of Probability in 1939, where the Bayes factor and its interpretive scale appear, and Bruno de Finetti supplied subjective-probability foundations from 1937. The practical revival came from computation - the Metropolis algorithm of 1953, the Hastings generalization in 1970, and the Gibbs sampling work of Alan Gelfand and Adrian Smith in 1990 - which finally made posteriors for realistic models tractable.