Thomas Bayes - Bayes and Bayesianism

Bayes and Bayesianism

Bayesian probability is the name given to several related interpretations of probability, which have in common the notion of probability as something like a partial belief, 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 random walk techniques. The use of the Bayes theorem has been extended in science and in other fields.

Bayes himself might not have embraced the broad interpretation now called Bayesian. It is difficult to assess Bayes' philosophical views on probability, since his essay does not go into questions of interpretation. There Bayes defines probability as follows (Definition 5).

The probability of any event is 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

In modern utility theory, expected utility can (with qualifications, because buying risk for small amounts or buying security for big amounts also happen) be taken as the probability of an event times the payoff received in case of that event. Rearranging that to solve for the probability, Bayes' definition results. As Stigler points out, 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' 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.