Two new properties of mathematical likelihood

Abstract

To Thomas Bayes must be given the credit of broaching the problem of using the concepts of mathematical probability in discussing problems of inductive inference, in which we argue from the particular to the general; or, in statistical phraselogy, argue form the sample to the population, from which, ex hypothesi, the sample was drawn. Bayes put forward, with considerable caution, a method by which such problems could be reduced to the form of problems of probability. His method of doing this depended essentially on postualting a priori knowledge, not of the particular population of which our observations from a sample, but of an imaginary population of populations from which this population was regarded as having been drawn at random. Clearly, if we have possession of such a priori knowledge, our problem is not properly an inductive one at all, for the population under discussion is then regarded merely as a particular case of a general type, of which we already possess exact knowledge, and are therefore in a positioin to draw exact deductive inferences. To the merit of broaching a fundamentally important problem, Bayes added that of perceiving, much more clearly than some of his followers have done, the logical weakness of the form of solutiion he put forward. Indeed we are told that it was his doubts respecting the validity of the postulate needed for establishing the method of inverse probability that led to this withholding his entire treatise from publication. Actually it was not published until after his death.