Statistical inference from an axiomatic viewpoint

Abstract

The introduction of the axioms of utility led many statisticians to think that there is only one correct statistical inference, that which complies with the results of decision theory. In reality, every theory comes to conclusions which require choices which in turn have unavoidable pragmatic consequences. But, since the axioms of a theory are accepted, it is self-sufficient. The truth of its results is relative to our conventions. We can discuss these conventions, but, unless the assumptions are changed from the very beginning, the solution to the problem of statistical inference must be found within probability calculus. On the other hand, if we accept de Finetti's coherence principle, then the usual assumptions of probability theory and the consequent rules of inference (Bayes' role included) appear inevitable. In this context, we agree with Lindley (1990), "according to this view, all manipulations in inference are solely and entirely within the calculus of probability. The mathematics is that of probability". However, following an axiomatic viewpoint, we go to a further conclusion and we agree with Carnap that statistical inference can be regarded as a simple inductive inference. The axiomatic approach to statistical inference has not yet received a sufficient attention in Statistics. Many prejudices remain in this area. In particular, the claims of superiority of testing and other decisional procedures are unjustified. An exhaustive analysis of the process of the updating of probabilities may allow us to correct the distortions due to an acritical application of theories in which the inferential situation is examined in a particular context. In line with these ideas, in this paper statistical inference is consistently developed from a pure Bayesian viewpoint. The approach is then illustrated by means of some numerical examples.