Editing Thomas Bayes (section)

== Bayesianism ==
[[Bayesian probability]] is the name given to several related interpretations of [[probability]] as an amount of epistemic confidence – the strength of beliefs, hypotheses etc. – rather than a frequency. This allows the application of probability to all sorts of propositions rather than just ones that come with a reference class. "Bayesian" has been used in this sense since about 1950. Since its rebirth in the 1950s, advancements in computing technology have allowed scientists from many disciplines to pair traditional Bayesian statistics with [[Markov chain Monte Carlo|random walk]] techniques. The use of the [[Bayes' theorem]] has been extended in science and in other fields.<ref>[[John Allen Paulos|Paulos, John Allen]]. [https://www.nytimes.com/2011/08/07/books/review/the-theory-that-would-not-die-by-sharon-bertsch-mcgrayne-book-review.html?_r=1&scp=1&sq=thomas%20bayes&st=cse  "The Mathematics of Changing Your Mind,"] ''New York Times'' (US). 5 August 2011; retrieved 6 August 2011</ref>

Bayes himself might not have embraced the broad interpretation now called Bayesian, which was in fact pioneered and popularised by [[Pierre-Simon Laplace]];<ref>Stigler, Stephen M. (1986) ''The history of statistics.'', Harvard University press. pp&nbsp;97–98, 131.</ref> it is difficult to assess Bayes's philosophical views on probability, since his essay does not go into questions of interpretation. There, Bayes defines ''probability'' of an event as "the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the value of the thing expected upon its happening" (Definition 5). In modern [[utility]] theory, the same definition would result by rearranging the definition of expected utility (the probability of an event times the payoff received in case of that event – including the special cases of buying risk for small amounts or buying security for big amounts) to solve for the probability. As Stigler points out,<ref name="stigler86history" /> this is a subjective definition, and does not require repeated events; however, it does require that the event in question be observable, for otherwise it could never be said to have "happened". Stigler argues that Bayes intended his results in a more limited way than modern Bayesians. Given Bayes's definition of probability, his result concerning the parameter of a [[binomial distribution]] makes sense only to the extent that one can bet on its observable consequences.

The philosophy of Bayesian statistics is at the core of almost every modern estimation approach that includes conditioned probabilities, such as sequential estimation, probabilistic machine learning techniques, risk assessment, simultaneous localization and mapping, regularization or information theory. The rigorous axiomatic framework for probability theory as a whole, however, was developed 200 years later during the early and middle 20th century, starting with insightful results in [[ergodic theory]] by [[Plancherel]] in 1913.{{Citation needed|date=April 2021}}