326 
ME, E. A. FISHEE ON THE MATHEMATICAL 
known as that of inverse probability. Thus, if tbe same observed result A might be 
the consequence of one or other of two hypothetical conditions X and Y, it is assumed 
that the probabilities of X and Y are in the same ratio as the probabilities of A occurring 
on the two assumptions, X is true, Y is true. This amounts to assuming that before 
A was observed, it was known that our universe had been selected at random for an 
infinite population in which X was true in one half, and Y true in the other half. 
Clearly such an assumption is entirely arbitrary, nor has any method been put forward 
by which such assumptions can be made even with consistent uniqueness. There 
is nothing to prevent an irrelevant distinction being drawn among the hypothetical 
conditions represented by X, so that we have to consider two hypothetical possibilities 
Xj and X,, on both of which A will occur with equal frequency. Such a distinction 
should make no difference whatever to our conclusions ; but on the principle of inverse 
probability it does so, for if previously the relative probabilities were reckoned to be 
in the ratio x to y, they must now be reckoned 2x to y. Nor has any criterion been 
suggested by which it is possible to separate such irrelevant distinctions from those 
which are relevant. 
There would be no need to emphasise the baseless character of the assumptions made 
under the titles of inverse probability and Bayes’ Theorem in view of the decisive 
criticism to which they have been exposed at the hands of Boole, Venn, and Chrystal, 
were it not for the fact that the older writers, such as Laplace and Poisson, who accepted 
these assumptions, also laid the foundations of the modern theory of statistics, and have 
introduced into their discussions of this subject ideas of a similar character. I must 
indeed plead guilty in my original statement of the Method of the Maximum Likeli¬ 
hood (9) to having based my argument upon the principle of inverse probability ; in the 
same paper, it is true, I emphasised the fact that such inverse probabilities were relative 
only. That is to say, that while we might speak of one value of p as having an inverse 
probability three times that of another value of p, we might on no account introduce 
the differential element dp, so as to be able to say that it was three times as probable 
that p should lie in one rather than the other of two equal elements. Upon considera¬ 
tion, therefore, I perceive that the word probability is wrongly used in such a connection : 
probability is a ratio of frequencies, and about the frequencies of such values we can 
know nothing whatever. We must return to the actual fact that one value of p, of 
the frequency of which we know nothing, would yield the observed result three times 
as frequently as would another value of p. If we need a word to characterise this 
relative property of different values of p, I suggest that we may speak without confusion 
of the likelihood of one value of p being thrice the likelihood of another, bearing always 
in mind that likelihood is not here used loosely as a synonym of probability, but simply 
to express the relative frequencies with which such values of the hypothetical quantity 
p would in fact yield the observed sample. 
The solution of the problems of calculating from a sample the parameters of the 
hypothetical population, which we have put forward in the method of maximum likeli- 
