B ayes' Theorem 



An Expository Presentation * 



By EDWARD C. MOLINA 



B AYES' theorem made its appearance as the ninth proposition in an 

 essay which occupies pages 370 to 418 of the Philosophical 

 Transactions, \^ol. 53, for 1763. An introductory letter written by 

 Richard Price, "Theologian, Statistician, Actuary and Political 

 Writer," ^ 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 "The Fundamental Problem of Practical Sta- 

 tistics." Opinions differ, however, as to the validity and significance 

 of the solution submitted in the essay for the problem in question. In 

 view of this situation I shall limit myself today to an exposition of the 

 fundamental characteristics of the problem Bayes' theorem deals with 

 and shall give no consideration to its interesting applications. 



The exposition may be outlined as follows: after specifying the class 

 of problems to which Bayes' theorem pertains I shall : 



* Read before the American Statistical Association during the meeting of the 

 American Association for the Advancement of Science in Cleveland, Ohio, December, 

 1930. 



1 These titles are associated with the name of Price in the frontispiece portrait of 

 him bound with the December, 1928, issue of Biometrika. 



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