Dazed and Confused

Bayes and Bayesianism
Bayesian probability is the name given to several related interpretations of probability, which have in common the application of probability to any kind of statement, not just those involving random variables. "Bayesian" has been used in this sense since about 1950.

It is not at all clear that Bayes himself would have embraced the very broad interpretation now called Bayesian. It is difficult to assess Bayes' philosophical views on probability, as the only direct evidence is his essay, which does not go into questions of interpretation. In the essay, 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 chance of the thing expected upon it's [sic] happening
In modern utility theory we would say that expected utility is the probability of an event times the payoff received in case of that event. Rearranging that to solve for the probability, we obtain Bayes' definition. As Stigler (citation below) 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". {Some would argue, however, that things can happen without being observable.}

Thus it can be argued, as Stigler does, that Bayes intended his results in a rather 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.


Modern applications
The search engine Google, and the information retrieval company Autonomy Systems, employ Bayesian principles to provide probable results to searches. Microsoft is reported as using Bayesian "probabalistic" mathematics in its future Notification Platform to filter unwanted messages.