Extended probabilities in Statistics

We propose a new, more general definition of extended probability measures. We study their properties and provide a behavioral interpretation. We use them 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 we progressively learn the true composition of $\Omega$. We describe how to update extended probabilities in this setting, and introduce the concept of lower extended probabilities. We provide two examples in the fields of ecology and opinion dynamics.