364 Prof. Donkiu on certain Questions relatiny to 



a person whose state of information (antecedently to the event E) 

 was such that it was to hiin the dejlnitire a priori probability of 

 H. Thus it is the expression (II.) which is right in principle, 

 though it happens^ remarkably, that (I.) is. i'ight in remit, ■AS'Wfe 

 shall see iimnediately. ' "" ^' "' -:it) pji: 



18. For it must be observed that both processes arc defective 

 in this respect, that they do not make use of the ol)served event E 

 to hnprove our provisional value of p ; that is, to get a better ap- 

 proximation to the true value of /», calculated antecedently to the 

 event. In order to do this we obser\e that (j>{p)dp expressing 

 our antecedent belief that the value in question lies between p 

 andjo + f^;, the antecedent probability that it does lie within 

 those limits mid that the event will happen is 



{p{a + a) + il-p){/j + ^)}<j>{p)dp; 



whence the probability after the event the true value of p lies 

 between p and p + dp is this expression divided by the sura of its 

 values for all values of y; (between and 1) separated by inter- 

 vals equal to (^p. The integration gives (17.) 



'57(« + «) + (l— C7)(6 + /3); 



and thus 



p{a + u) + {l-p)ib + ^) 

 . vi... „-n ^. ^^a + u) + {l-vr){b + ^f^P>''^',. ■nUAl}^'! 



is the eXpfesion of our belief, influenced by the ev&nr, thstt' 'the 

 value of p, calculated before the event, would have turned out to 

 lie between p and p + dp. We might from this expression obtain 

 a new provisional value of p corrected by the event. Bat if we 

 proceeded to substitute this for yj in (P.), art. 16, we should again 

 commit the fallacy of treating a provisional value as if it were 

 definitive. Instead of this, therefore, we take the product of (4>.) 

 and (P.), namely, 



{pa + {l-p)b)4>[lJ)dp 

 t!r(« + «) + (l-^)(Z- + /8)' 



which is the expression of our belief, after the event, that the 

 true value of jl», antecedently to the event, was between p and 

 p-\-dp, and i\\zi the cause C exists; and integrating this from 

 p-=Q to p=^\, we get the expression of our whole a posteriori 

 belief that C exists, namely, 



'Sja-\-{\—m)b km + l 



- . i . QY • 



sr(a + a) + (l— t7)(6 + /S) -mJ^m' 



and this is the best solution of which the problem admits, so 

 long as we do not calculate the definitive value oi p. Moreover, 

 since it is obtained by taking the sum of the values of (P.), each 



