Bayes' Theorium - An Expository Presentation

AYES' theorem made its appearance as the ninth proposition in an essay which occupies pages 370 to 418 of the Philosophical Transactions, Vol. 5.3, for 1763. An introductory letter written by Richard Price, "Theologian, Statistician, Actuary and Political Writer," 1 begins thus: " I now send you an essay which I have found amongst the papers of our deceased friend Mr. Bayes, and which, in my opinion, has great merit, and well deserves to be preserved." A few lines farther on Price says: " In an introduction which he has writ to this Essay, he says, that his design at first in thinking on the subject of it was, to find out a method by which we might judge concerning the probability that an event has to happen, in given circumstances, upon supposition that we know nothing concerning it but that, under the same circumstances, it has happened a certain number of times, and failed a certain other number of times." " Every judicious person will be sensible that the problem now mentioned is by no means merely a curious speculation in the doctrine of chances, but necessary to be solved in order to a sure foundation for all our reasonings concerning past facts, and what is likely to be hereafter." No one will dispute the importance ascribed to Bayes' problem by Price; in fact, a paper by Karl Pearson on an extension of Bayes' problem is entitled " T h e Fundamental Problem of Practical Statistics." Opinions differ, however, as to the validity and significance of the solution submitted in the essay for the problem in question.