Prior Distributions From Pseudo-Likelihoods in the Presence of Nuisance Parameters

Abstract

Consider a model parameterized by θ == ( ψ , λ ), where ψ is the parameter of interest. The problem of eliminating the nuisance parameter λ can be tackled by resorting to a pseudo-likelihood function L **( ψ ) for ψ —namely, a function of ψ only and the data y with properties similar to those of a likelihood function. If one treats L **( ψ ) as a true likelihood, the posterior distribution π **( ψ | y ) ∝ π ( ψ ) L **( ψ ) for ψ can be considered, where π ( ψ ) is a prior distribution on ψ . The goal of this article is to construct probability matching priors for a scalar parameter of interest only (i.e., priors for which Bayesian and frequentist inference agree to some order of approximation) to be used in π **( ψ || y ). When L **( ψ ) is a marginal, a conditional, or a modification of the profile likelihood, we show that π ( ψ ) is simply proportional to the square root of the inverse of the asymptotic variance of the pseudo-maximum likelihood estimator. The proposed priors are compared with the reference or Jeffreys’ priors in four examples.