Short answers, pulled from the story.
Bayesian inference is a method of statistical inference that uses Bayes' theorem to calculate the probability of a hypothesis given prior evidence, then updates that probability as new information arrives. It starts with a prior probability, multiplies it by a likelihood derived from observed data, and produces a posterior probability. The posterior then serves as the prior for the next round of evidence.
The method is named after Thomas Bayes (1701-1761), who proved that probabilistic limits could be placed on an unknown event. Pierre-Simon Laplace (1749-1827) independently introduced what is now called Bayes' theorem and applied it to problems in celestial mechanics, medical statistics, reliability, and jurisprudence.
Cromwell's rule states that if a prior probability is set to exactly zero or exactly one, no amount of evidence can ever change it. It is a direct logical consequence of Bayes' theorem, and it warns against assigning absolute certainty or impossibility to any hypothesis before data is observed.
Bayesian methods experienced dramatic growth in research and applications in the 1980s, mostly attributed to the discovery of Markov chain Monte Carlo methods, which removed many of the previous computational barriers. Interest further expanded with the rise of machine learning and complex nonstandard applications.
Bayesian inference is used to estimate the probability that an email is spam by treating each word as evidence that updates the hypothesis. Applications built on this approach include CRM114, DSPAM, Bogofilter, SpamAssassin, SpamBayes, Mozilla, and XEAMS. Bayesian pattern recognition techniques of this kind date to the late 1950s.
Yes. In the case R v Adams, a defense expert witness explained Bayes' theorem to a jury in the United Kingdom. The Court of Appeal upheld the resulting conviction but stated that introducing Bayes' theorem into a criminal trial plunges the jury into inappropriate and unnecessary realms of theory and complexity, deflecting them from their proper task.