Importance of the prior mass for agreement between frequentist and Bayesian approaches in the two-sided test

摘要

A Bayesian test for H_o : θ = θ_o versus H_1 :θ≠θ_0 is developed. The methodology consists of fixing a sphere of radius δ around θ_0, assigning to Ho a prior mass, π_0, computed by integrating a density function n(θ) over this sphere, and spreading the remainder, 1 - π_0, over H_1] according to π(θ). The ultimate goal is to show when p values and posterior probabilities can give rise to the same decision in the following sense. For a fixed level of significance α, when do e_1 ≤e_2 exist such that, regardless of the data, a Bayesian proponent who uses the proposed mixed prior with π_0 ∈ (e_1,e_2) reaches, by comparing the posterior probability of H_0 with 1/2, the same conclusion as a frequentist who uses α to quantify the p value? A theorem that provides the required constructions of e_1 and e_2 under verification of a sufficient condition (e_1≤e_2) is proved. Some examples are revisited.