Extended probabilities and their application to statistical inference

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).