ABSTRACT
The problem stated in Bayes’ famous paper (Figure 1.1) involves two key ingredients. One is the use of probability as a means of expressing uncertainty about an unknown quantity of interest. The other is the conditional nature of the problem: what Bayes was interested in evaluating was the conditional probability of failure in a single trial, given some data on the previous number of failures. Put another way, he wanted to learn about the failure probability on the basis of observed data. In modern language, this translates to requiring p(θ|y, n) where θ is the unknown failure probability and we have observed data on y failures out of n binomial trials. Bayes proposed a theorem (easily provable from the axioms of probability) relating conditional and marginal probabilities of random variables which he used to calculate the required conditional probability for his problem.
