Bayesianism: Its Unifying Role for Both the Foundations and Applications of Statistics

A fruitful discussion on the subject requires, first of all, to disentangle it from the difficulties arising whenever single aspects are considered separately, as unconnected technicalities concerning isolated problems. Induction is viewed usually in many very different, partial, unsatisfactory, isolated fragments of theories. There is the formalistic one of a kind of logicians; there are the two mathematical ones concerned respectively with inductive reasoning and inductive behaviour; and all became more and more complex owing to a growing inflation of formalism. Formalism is, in effect, the illusory remedy to the insufficiencies arising from isolation, and is in turn a factor of enhancing isolation. The thesis of the present paper goes just in the opposite direction, trying to show, as clearly and simply as possible that, avoiding the misleading preconceptions of artificial unilateral constructions, the whole subject admits a unique and very natural interpretation and a simple universal answer. That is the Bayesian theory: but it is a pity it is called a "theory" and has a name, for the same reasons that led Cornfield to say (noting that Bayes' theorem is but an obvious result) that "it is overly solemn to call it a theorem at all". Probabilities have a unique true meaning as degrees of belief (a subjective one, although one must take into reasonable account the objective data available: symmetries, frequencies, analogies, from all his experience), and must coherently agree together and coherently evolve by changing of the state of information. (That is, summarizing, the Bayes' theorem.) All that is imposed, in a unique way, by concordant reasons pertaining to each of the aspects (and are, in their essence, an unique reason which presents itself under slightly modified appearance in the various occurrences). These views, which in the present summary could only be sketched in an abstract and apodictical form, are carefully exposed in the present paper, comparing the effect of conformity to them, or of deviations, on all possible facets of the problems concerning induction, as for logical conclusions and for inductive reasoning and behaviour. Objections against subjectivity should be overcome; abstaining from subjective opinions yields in fact no improvement on objectivity; on the contrary, a subjectivistic integration to the barren bulk of the objective data appears to be necessary, so that, if it is missing, we are led not to any better situation but to a more erratic one.