The randomness of the inferred parameters. A machine learning framework for computing confidence regions.

We start from the very operational perspective - having data, organize them in a suitable way to be used in the future - to enter the long standing fray on the nature of inferred parameters within a machine learning thread. Still in an operational perspective, we introduce a parametric inference approach that unprecedentedly gets rid of most drawbacks incurred by current methods to compute confidence intervals. The key idea is to consider the parameters of the distribution underlying a sample to be random, where randomness is expressed in terms of a probability measure of the compatibility of the parameter values with the actually observed data. The probability is understood, in a frequentist acceptation, in terms of the asymptotic frequency of those parameter values matching the observed sample in a story of infinite observations. The aim of this paper is to recap and complete theoretical results obtained through our approach as presented in preceding papers. In particular, here we focus on statistical tools both for computing confidence regions, at the basis of appraising the learnability of a function, and for checking their efficacy. We basically support our theory with a series of well-known benchmarks where, as for both volume and coverage of the confidence regions, our method proves superior - with very few ties - to those of competitors. Then we mention some results in computational learning theory that have been achieved recently exactly by adopting our approach, with a special focus on a new data_ accuracy - sample_complexity trade off.