H. E. SoPER, A. W. Young, B. M. Cave, A. Lee, K. Pearson 359 
Statistical workers cannot be too often reminded that there is no vahdity in 
a mathematical theory pure and simple. Bayes' Theorem must be based on 
experience, the experience that where we are it 'priori in ignorance all values are 
equally likely to occur. This is not the case in the present illustration, and we 
must use our past experience in the same way as we should use our past experience of 
equal frequency ; the appeal to this experience has here absolutely the same validity 
as in Bayes' case and cannot be for a moment neglected. We see that our new 
experience scarcely modifies the old and this is what we should naturally conjecture 
would be the case. If we increase the size of the new sample, then ultimately 
Ijm {n — 1) becomes very small, and we approach nearer the value -59194 given 
by Bayes' Theorem. But past experience will bias the value obtained from the 
new material for a long time, and we see that according to the value of the past 
experience p may vary from -46225 to -59194. It will thus be evident that in 
problems like the present the indiscriminate use of Bayes' Theorem is to be 
deprecated. It has unfortunately been made into a fetish by certain purely 
mathematical writers on the theory of probabihty, who have not adequately 
appreciated the limits of Edgeworth's justification of the theorem by appeal to 
general experience. 
Case (iii). Past Experience a Factor, but not the Dominating Factor of Judgment. 
Cases can arise in which p = p is not a very close approximation, i.e. when we 
have some past experience, but not a very concentrated one of like correlations. 
In this case we must return to Equation (Ixiii), and we shall assume pr = p^y^ + e, 
where p^^ is some fairly close approximation to pr. We shall write pr = pQ^. We 
find 
l^^^)^n ]Po + ,n{n-l)(r^-p,^)\^"-' 
+ m{n-l)ir^-~p7)l ^""^ + |(» + 1) Pa + - _ -^4) | A< " (' - Po ) 
(Ixxii), 
where n {1 - p^*) 1^+^ = {2n - 1) po^4 + {n - 1) 4-i (Ixxiii), 
equations which caji be readily expressed in terms of ^'s. 
Unfortunately the approximation obtained by equating the numerator of e 
to zero and using (Ixxiii) as simultaneous equations is not very rapidly obtained 
as the resulting equation is now of the eighth order. It is better from the data 
themselves to guess a reasonable value for p^^ and start the approximation from 
this. 
Illustration. The correlation between the maximum length and breadth of 
crania is not very definitely known. Its mean is about -30, but the values deter- 
mined for it range from nearly zero to -6. Assuming the standard deviation to 
be -1, what is the "most likely value" to give to this correlation in the case of a 
sample of 25 skulls showing a correlation of -50 ? 
