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placed ball and a "failure" that it should be greater. The chance therefore of ^ successes and q 
failures now happening is 
\'P+q 
\P\l 
It is solely the fact that all possible values of the variate x are made a /)?7bn equally likely 
that makes the chance of a success x^ equal to the variate itself. Those who criticise Bayes after 
reading his actual paper, say that he ought not to have made the chances of a ball being placed 
anywhere on the table equally likely. He makes in fact his distribution of the variate x a straight 
line — a somewhat unusual form of frequency distribution*. My answer to that objection to Bayes' 
work was that you can make the distribution of that variate — i.e. position on the table — any 
continuous curve you please as Bayes' Theorem with Bayes' results will flow from it equally well. 
Against this position my critics raise the cry that the chance is no longer x of a success and \ —x 
of a failure. Of course not, because that depends on horizontality of frequency distribution and 
it was merely fortuitous that for that case Bayes' variate x corresponded to a chance. In other 
cases the chance is a function of the variate x and not x itself. But if the critics say : Then this is 
not what we mean by Bayes' Theorem, I would reply : Quite so, but it is what Bayes meant by 
his own Theorem, and it probably fits much better the type of cases to which we are accustomed 
to apply it than what you mean by Bayes' Theorem. 
Let me illustrate this point. An event has happened p times and failed q times ; what is the 
chance that in r + s further trials it will occur r times and fail si. The critics say: this is our 
Bayes' Theorem, not your Bayes' Theorem. I reply that it is both, but that your way of solving 
it is not Bayes'. Perhaps Bayes saw further than some of the critics who have not troubled to 
read his original paper. What Bayes said was this, the event will happen when there is an excess 
(or it may be defect) of a certain variate, but I do not know what is the hmiting value of this 
variate a priori. Look at Bayes' billiard table from a more modern standpoint. Men will sicken 
from a disease when their resistance falls below x a.n a priori unknown value of the variate. 
Bayes took the chance of this limiting value lying between x and x+dx to be dx, if the total range 
be taken as unity. He ought to have taken it cj> (x) dx, where (p (x) is arbitrary. He took the 
chance of occurreuce of the disease to be x, when he ought to have taken it I <p (x) dx = Pj.; and 
Jo 
of failure I —x instead of I — P^. 
But had he taken these better values he would have reached finally precisely the .same result 
as he did by his equal distribution of ignorance. Bayes made every value of his variate x equally 
likely. He ought to have given them a perfectly arbitrary frequency distribution. A priori all 
degrees of immunity are not equally likely to be the limiting value in the case of a disease. The 
generalised Bayes as thus envisaged has a very wide application to vital statistics ; in fact it seems 
to me to entirely replace the other sort of Bayes' Theorem suggested by his critics. 
Nay I would go further, and say that it is Bayes' Theorem in Bayes' sense that we need in 
most questions of prediction of the future from the past. If, for example, two men play a set of 
games and A has won p and B q games and we consider the chance in the following r + s games of 
A's winning r and B's winning s, then I believe that A's winning may be accurately considered as 
depending on the excess of a certain variate x which is a function of A's skill relative to B's. 
A priori we do not know what the value of x is for which A will win. All we can say is that when 
relative to B he exhibits a certain excess of skill he will win, but that we must not assume with 
Bayes that all limit-values of x are equally likely. We must take any frequency curve for the 
possible distribution of x. 
I believe that in most cases such a variate may be hypothecated and if it can the objection to 
Bayes that he made all positions of his balls on the table " equally likely " can be removed, and 
if removed one fundamental objection to his theorem as he stated it, i.e. in terms of excess or 
defect of a variate, disappears. K. P. 
* It is in fact the "rectangle point" )3, = 0, just as limited a distribution as the Gaussian 
^1 = 0, ^j = 3. 
