the Theory of Probabilities. 365 



multiplied respectively by the quantity of our actual belief that 

 it is the right value, the above expression may also be called our 

 expectation oi the, xdilwc oi (P.). And since we see, as before 

 remarked, that it coincides with the expression (I.), art. 17, we 

 get this theorem; that our expectation of the value of (P.), cor- 

 rected by the influence of the event upon the form of the func- 

 tion ^, is the same as the definitive value of (P.) would be if our 

 uncorrected expectation ot of the value of pj were its definitive 

 value. This theorem, which could not, I conceive, be assumed 

 a priori, might be used to abridge many processes ; and it might 

 probably be shown that an analogous theorem subsists for every 

 problem involving a priori probabilities among its data. 



19. In order, however, to illustrate still further the subject 

 under discussion, let us return to the suppositions of art. 16, 

 and inquire what is the a posteriori value of p ; in other words, 

 what is our belief, after the event, that the hypothesis H is true. 

 If we represent by p, as before, the definitive a prioi-i value of 

 the probability that H is true, then the ci posterio?i probability 

 in question is easily seen to be 



pift + «) 



;;(« + «) + (l-io)(* + /3)' 



but if we do not possess a definitive value of p, we may apply 

 exactly the same reasoning as in the former case, and the solution 

 of the problem will be obtained by substituting w for p in the 

 expression just written. I have introduced this addition to the 

 problem of art. 16, for the sake of pointing out the fallacy of a 

 process which might, at first sight, appear as legitimate as that 

 which has just been employed. It might seem, namely, that 

 we ai'c at liberty to introduce in the expression (P.), art. 16, the 

 value obtained in this article for the probability of H, as an im- 

 proved value of p. But this would be to confound an o priori 

 with an a jjosteriori probability, and would obviously be arguing 

 in a circle. (In fact, if the substitution were allowable at all, it 

 might be made successively an indefinite number of times, which 

 is easily shown to lead to an absurd result.) It confounds our 

 belief, after the event, of the truth ofW, Avith our belief, after 

 the event, of mhat vwuld have been our fjelief of H before the 

 event, if certain calculations had been completely made which 

 have not been coni])letely made. These are two totally different 

 probaljilities, though botli are affected by the event ; the former 

 necessarily, the latter only because of the defect of calculation. 



20. I trust I may be excused for devoting so much space to 

 the discussifjn of j)rocesses so well known, on the ground that 

 there may possibly be readers to whom processes are more fami- 

 liar tliau pruiciples ; and that so long as any shadow of obscu- 



