On the theory of errors and least squares

Abstract

1. In my “Scientific Inference,” chapter V, I found that the usual presentation of the theory of errors of observation needed some modification, even where the probability of error is distributed according to the normal law. One change made was in the distribution of the prior probability of the precision constant h . Whereas this is usually taken as uniform (or ignored), I considered it better to assume that the prior probability that the constant lies in a range dh is proportional to dh/h . This is equivalent to assuming that if h 1 / h 2 = h 3 / h 4 , h is as likely to lie between h 1 and h 2 as between h 3 and h 4 ; this was thought to be the best way of expressing the condition that there is no previous know­ ledge of the magnitude of the errors. The relation must break down for very small h , comparable with the reciprocal of the whole length of the scale used, and for large h comparable with the reciprocal of the step of the scale; but for the range of practically admissible values it appeared to be the most plausible distribution. The argument for this law can now be expressed in an alternative form. The normal law of error is supposed to hold, but the true value x and the pre­cision constant h are unknown. Two measures are made: what is the pro ability that the third observation will lie between them ? The answer is easily seen to be one-third.