"Additivity" versus "Maxitivity" at the heart of the paradoxical and efficient nature of Statistics

TL;DR

This paper explores the fundamental differences between additive and maxitive support measures in statistical inference, highlighting their limitations and proposing a combined approach as a practical solution.

Contribution

It demonstrates the mutual exclusivity of additive and maxitive principles in statistical theory and advocates for a hybrid method to address partial information and ignorance.

Findings

Additive and maxitive support measures are mutually exclusive.

Neither approach fully handles ignorance or partial data.

A combined approach offers a practical middle ground.

Abstract

Unlike the Probability Theory based on additivity, Statistical Inference seems to hesitate between "Additivity" and a so-called "Maxitivity" approach. After a brief overview of three types of principles for any (parametric) statistical theory and the proof that these principles are mutually exclusive, the paper shows that two kinds of support measures are conceivable, an additive one and a maxitive one (based on maximization operators). Unfortunately, none of them is able to cope with the ignorance part of the statistical experiment and, in the meantime, with the partial information given through the structure of the data. To conclude, the author promotes the combined use of both approaches, as an efficient middle-of-the-road position for the statistician.