Uniform and subuniform posterior robustness: The sample size problem

Abstract

The following general question is addressed: given i.i.d. realizations X1,X2,…,Xn from a distribution Pθ with parameter θ, where θ has a prior distribution π belonging to some family Γ, is it possible to prescribe a sample size n0 such that, for n⩾n0, obtaining posterior robustness is guaranteed for any actual data we are likely to see or even for all possible data. Formally, we identify a ‘natural’ set C such that P (the observation vector X∉C)⩽ε, for all possible marginal distributions implied by Γ, and protect ourselves for all X in the set C. Typically, such a set C exists if Γ is tight. There are two aspects in these results: one of them is establishing the plausibility itself; this is done by showing uniform convergence to zero of ranges of posterior quantities. This part forms the mathematical foundation of the program. The second aspect is providing actual sample size prescriptions for a specific goal to be attained. This forms the application part of the program. In this article, we only consider testing and set estimation problems relating to the normal distribution with conjugate priors.