Higher Order Imprecise Probabilities and 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.