280 BELL SYSTEM TECHNICAL JOURNAL 



IV 



Equation (5) is a very beautiful formula; Init we must be cautious. 

 More than one high authority has insinuated that its beauty is only 

 skin deep. Speaking of Laplace's generalization and extension of the 

 theorem, George Chrystal, the English mathematician and actuary, 

 closed a severe attack on the whole theory of a posteriori probability ^ 

 with the statement that "Practical people like the Actuaries, however 

 much they may justly respect Laplace, should not air his weaknesses 

 in their annual examinations. The indiscretions of great men should 

 be quietly allowed to be forgotten." 



Chrystal's advice as to the attitude one should assume toward "the 

 indiscretions of great men" is excellent, but in the case under con- 

 sideration, it was the plaintiff rather than the defendant who com- 

 mitted indiscretions; this is discussed in a paper by E. T. Whittaker * 

 entitled "On Some Disputed Questions of Probability." 



The discussions and disputes, which began shortly after the birth of 

 the formula in 1763 and which have not as yet subsided, may be 

 divided into two classes: 



\. Discussions concerning problems in which it is known that the a 



priori existence function is not a constant. 

 2. Discussions concerning problems in which nothing whatever is 



known concerning the a priori existence function. 



The discussions of Class 1 are out of order in so far as Bayes' theorem 

 is concerned; recourse should be had to formula (4), Laplace's generali- 

 zation of the Bayes' theorem, when it is known that w{x) is not a 

 constant. Failure to differentiate explicitly between equations (4) 

 and (5) has created a great deal of confusion of thought concerning the 

 probability of causes. The discussions of Class 2 have centered on 

 what Boole called "the equal distribution of our knowledge, or rather 

 of our ignorance," that is to say "the assigning to different states of 

 things of which we know nothing, and upon the very ground that we 

 know nothing, equal degrees of probability." Regarding the legiti- 

 macy of this procedure Bayes himself contributed a very important 

 scholium which appeared in his essay on pages 392 and 393. The 

 argument in this scholium, based on a corollary to Proposition 8 of the 

 essay, may be summarized as follows: 



Assuming that all values of x are a priori equally likely and that the 

 N throws of a ball on the table have not yet been made, the probability 



■• "Oil Some P'undamental Principles in the Theory of ProbabiHty," Transactions 

 oj the Actuarial Society of Edinburgh, Vol. 11, No. 13. 



^ Transactions of the Faculty of Actuaries in Scotland, \o\. \TII, Session 1919-1920. 



