FOUNDATIONS OF THEORETICAL STATISTICS. 
327 
hood, consists, then, simply of choosing such values of these parameters as have the 
maximum likelihood. Formally, therefore, it resembles the calculation of the mode of 
an inverse frequency distribution. This resemblance is quite superficial : if the scale 
of measurement of the hypothetical quantity be altered, the mode must change its 
position, and can be brought to have any value, by an appropriate change of scale ; but 
the optimum, as the position of maximum likelihood may be called, is entirely unchanged 
by any such transformation. Likelihood also differs from probability* in that it is not 
a differential element, and is incapable of being integrated : it is assigned to a particular 
point of the range of variation, not to a particular element of it. There is therefore an 
absolute measure of probability in that the unit is chosen so as to make all the elementary 
probabilities add up to unity. There is no such absolute measure of likelihood. It 
may be convenient to assign the value unity to the maximum value, and to measure 
other likelihoods by comparison, but there will then be an infinite number of values 
whose likelihood is greater than one-half. The sum of the likelihoods of admissible 
values will always be infinite. 
Our interpretation of Bayes’ problem, then, is that the likelihood of any value of p 
is proportional to 
p x {l-p) y , 
and is therefore a maximum when 
x 
which is the best value obtainable from the sample ; we shall term this the optimum 
value of p. Other values of p for which the likelihood is not much less cannot, however, 
be deemed unlikely values for the true value of p. We do not, and cannot, know, from 
the information supplied by a sample, anything about the probability that p should lie 
between any named values. 
The reliance to be placed on such a result must depend upon the frequency distribution 
of x, in different samples from the same population. This is a perfectly objective 
statistical problem, of the kind we have called problems of distribution ; it is, however, 
capable of an approximate solution, directly from the mathematical form of the 
likelihood. 
When for large samples the distribution of any statistic, 6 } , tends to normality, we 
* It should be remarked that likelihood, as above defined, is not only fundamentally distinct from 
mathematical probability, but also from the logical “ probability ” by which Mr. Keynes (21) has recently 
attempted to develop a method of treatment of uncertain inference, applicable to those cases where we 
lack the statistical information necessary for the application of mathematical probability. Although, in 
an important class of cases, the likelihood may be held to measure the degree of our rational belief in a 
conclusion, in the same sense as Mr. Keynes’ “ probability,” yet since the latter quantity is constrained, 
somewhat arbitrarily, to obey the addition theorem of mathematical probability, the likelihood is a 
quantity which falls definitely outside its scope. 
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