Higher Order Imprecise Probabilities and Statistical Testing

TL;DR

This paper extends imprecise probability models to include higher order credal sets, addressing issues like Dilation and Belief Inertia, and explores their convergence properties with implications for statistical testing.

Contribution

It introduces higher order credal sets for imprecise probabilities and analyzes their updating rules and convergence behavior, providing new insights into prior selection for hypothesis testing.

Findings

Higher order credal sets can address standard issues in imprecise probabilities.

Finite simulations support the convergence of higher order credal sets to a uniform distribution.

The total-variation-uniform distribution emerges as a natural prior for statistical testing.

Abstract

We generalize standard credal set models for imprecise probabilities to include higher order credal sets -- confidences about confidences. In doing so, we specify how an agent's higher order confidences (credal sets) update upon observing an event. Our model begins to address standard issues with imprecise probability models, like Dilation and Belief Inertia. We conjecture that when higher order credal sets contain all possible probability functions, then in the limiting case the highest order confidences converge to form a uniform distribution over the first order credal set, where we define uniformity in terms of the statistical distance metric (total variation distance). Finite simulation supports the conjecture. We further suggest that this convergence presents the total-variation-uniform distribution as a natural, privileged prior for statistical hypothesis testing.