Noninformative Priors and Nuisance Parameters

Abstract

We study the conflict between priors that are noninformative for a parameter of interest versus priors that are noninformative for the whole parameter. Our investigation leads us to maximize a functional that has two terms: an asymptotic approximation to a standardized expected Kullback-Leibler distance between the marginal prior and marginal posterior for a parameter of interest, and a penalty term measuring the distance of the prior from the Jeffreys prior. A positive constant multiplying the second terms determines the tradeoff between noninformativity for the parameter of interest and noninformativity for the entire parameter. As the constant increases, the prior tends to the Jeffreys prior. When the constant tends to 0, the prior becomes degenerate except in special cases. This prior does not have a closed-form solution, but we present a simple, numerical algorithm for finding the prior. We compare this prior to the Berger-Bernardo prior.

Key Words:

• Asymptotic information
• Reference prior
• Tradeoff prior