362 Prof. Donkin on certain Questions relatiiig to 



like manner, if I am able to assign an approximate value p^ oi p, 

 with a belief, as to the precision of my approximation, expressed 

 by (f){p) ; where (j>ip) is a maximum for p^po, and <p{p)dp is 

 my belief that the true value would turn out to lie between p 



and p + dp, 4>{p) satisfying of course the condition /" <^[p)dp = \ ; 

 then the belief which I actually give to the truth of the hypo- 

 thesis is f p(f){p)dp. This might he called its provisional proba- 



bility. It expresses also, according to a well-known use of the 

 term, my expectation of the value of j) (as distinguished from its 

 most probable value, which is my original approximation p^. 

 Similar considerations apply to the second case mentioned in 

 art. 13, which it would be useless to repeat in detail, after what 

 has just been said. 



15. Let us, for distinction, call the " definitive ^^ value of a 

 probability, relative to a pai'ticular state of information, that 

 value which would be obtained if all the necessary calculations 

 were actually performed ; while any value adopted, as explained 

 in the last article, without the performance of these calculations, 

 may be called a " provisional " value. It is obvious that the 

 " definitive " value of an a priori probability cannot be altered 

 by any fresh information, or even by the discoveiy of the truth 

 or falsehood of the hypothesis to which it refers. For the new 

 probability so obtained has reference to a new state of information, 

 and has nothing to do with the qiiantity of belief appropriate to 



■ the former state. But the provisional value may be altered by 

 new information ; that is, the knowledge which we gain in a new 

 state of information may alter our estimate of what ivould have 

 turned out to be tlie quantity of belief appropriate to a former 

 state, if the calculation had been performed. And it is to be 

 remembered that the employment of new information to improve 

 our provisional estimate of the a jjriori probability of a hypothesis, 

 is a totally different thing from the employment of it to obtain 

 an a posteriori probability of the same hypothesis. These di- 

 stinctions arc very important ; and as I am not sure that they 

 are so generally understood as it appears to me that they ought 

 to be, I shall illustrate them by an example which will suffi- 

 ciently represent any problem of the class now considered. 



16. An event E has been observed, which can only have re- 

 sulted from some one or other of the causes C, C, . . . of which 

 any one would necessarily produce it, and no two coidd coexist. 

 It is required to assign the probability that it has resulted from 

 C ; it being known that the jn'obability (before the event) that 

 C existed, would be « if a certain hypothesis H were ti-ue, and b 

 if it were false ; whilst the probability that some one of the other 

 causes existed would be a if H were true, and /3 if it were false. 

 Let p be the a priori probability of H ; then the required a pos- 



