When is Conflict Normal?

Abstract

When data conflicts with quantified prior beliefs, seemingly small changes in the functional form of the prior and/or likelihood can have a profound effect on posterior inferences. This article provides results on asymptotic forms of the posterior when two information sources conflict. In particular, let x be from the likelihood p ₂(x − θ) with an unknown location parameter θ, with p ₁(θ) the prior on θ. Sufficient conditions on p ₁ and p ₂ are provided to ensure that as x → ∞ the posterior is asymptotically normal. The conditions cover all combinations in which p ₁ and p ₂ are proportional to exp { − |θ − x _i |^{τ _i} , with τ _i > 1. Conditions are also provided that ensure that as some of the data become extreme, the posterior variance goes to 0, converges to a constant, or diverges to infinity. Other asymptotic behavior involving conflicting information sources is also discussed.

Key Words:

• Bayesian rejection
• Box-Tiao densities
• Outliers
• Posterior normality
• Robustness