Extended probabilities and their application to statistical inference

TL;DR

This paper introduces a new class of extended probability measures, explores their properties and interpretations, and applies them to statistical inference problems involving unknown sample spaces, including species sampling and opinion dynamics.

Contribution

It develops a generalized framework for extended probabilities and demonstrates their use in inference when the underlying probability space is unknown.

Findings

Extended probabilities can be updated as more information about the sample space is obtained.

The concept of lower extended probabilities provides bounds in uncertain environments.

Application to species sampling and boomerang effect illustrates practical utility.

Abstract

We propose a new, more general definition of extended probability measures. We study their properties and provide a behavioral interpretation. We put them to use in an inference procedure, whose environment is canonically represented by the probability space $(\Omega,\mathcal{F},P)$, when both $P$ and the composition of $\Omega$ are unknown. We develop an ex ante analysis -- taking place before the statistical analysis requiring knowledge of $\Omega$ -- in which the true composition of $\Omega$ is progressively learned. We describe how to update extended probabilities in this setting, and introduce the concept of lower extended probabilities. We apply our findings to a species sampling problem and to the study of the boomerang effect (the empirical observation that sometimes persuasion yields the opposite effect: the persuaded agent moves their opinion away from the opinion of the persuading agent).