FOUNDATIONS OF THEORETICAL STATISTICS. 
325 
The postulate would, if true, be of great importance in bringing an immense variety 
of questions within the domain of probability. It is, however, evidently extremely arbi¬ 
trary. Apart from evolving a vitally important piece of knowledge, that of the exact 
form of the distribution of values of p, out of an assumption of complete ignorance, it is 
not even a unique solution. For we might never have happened to direct our attention 
to the particular quantity p : we might equally have measured probability upon an 
entirely different scale. If, for instance, 
sin 0 -- 2p — 1, 
the quantity, 6, measures the degree of probability, just as well as p, and is even, for 
some purposes, the more suitable variable. The chance of obtaining a sample of x 
successes and y failures is now 
( 1+sin o) x {i-sm e)y- 
applying the method of maximum likelihood, 
S (log f ) — x log (l +sin 0) + y log (l — sin 6) —n log 2, 
and differentiating with respect to 0, 
x cos 0 _ y cos 6 
1 +sin 0 1— sin 0 
whence 
sin 6 = 
x — y 
2n 
an exactly equivalent solution to that obtained using the variable p. But what a prion 
assumption are we to make as to the distribution of 0 ? Are we to assume that 0 is 
equally likely to lie in all equal ranges d0 ? In this case the a priori probability will 
be d0j 7 t, and that after making the observations will be proportional to 
(1 +sin 0) c (l — sin 0) y d0. 
But if we interpret this in terms of p, we obtain 
P 
(i -p)‘ 
dp 
\ZP{ 1 ~P) 
= p 2 ~^(l— pY " dp, 
a result inconsistent with that obtained previously. In fact, the distribution previously 
assumed for p was equivalent to assuming the special distribution for O, 
d f=—-de, 
the arbitrariness of which is fully apparent when we use any variable other than p. 
In a less obtrusive form the same species of arbitrary assumption underlies the method 
VOL. CCXXIT. — A. 2 7 , 
