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

Abstract

A Bayesian test for H ₀: θ=θ ₀ versus H ₁: θ≠θ ₀ is developed. The methodology consists of fixing a sphere of radius δ around θ ₀, assigning to H ₀ a prior mass, π₀, computed by integrating a density function π(θ) over this sphere, and spreading the remainder, 1−π₀, 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 ℓ₁≤ℓ₂ exist such that, regardless of the data, a Bayesian proponent who uses the proposed mixed prior with π₀∈(ℓ₁, ℓ₂) reaches, by comparing the posterior probability of H ₀ with ½, the same conclusion as a frequentist who uses α to quantify the p value? A theorem that provides the required constructions of ℓ₁ and ℓ₂ under verification of a sufficient condition (ℓ₁≤ℓ₂) is proved. Some examples are revisited.

Keywords:

• posterior probability
• multivariate point null hypothesis
• p-value
• Lindley's paradox

2000 Mathematics Subject Classification :

• 62F15
• 62F03