Default priors from pseudo-likelihoods in the presence of nuisance parameter

Consider a sampling parametric model with parameter θ = (ψ, λ), where ψ is the parameter of interest. Also in the Bayesian framework, an approach to the elimination of nuisance parameters is to use an appropriate pseudo-likelihood function LPS(ψ) for the parameter of interest only, in alternative to an integrated likelihood function. The goal of this paper is to select a class of default priors πPS(ψ) for a parameter of interest using pseudo-likelihood functions. Developing Stein’s (1985) and Tibshirani’s (1989) results, our approach is to require that the resulting pseudo-posterior intervals, based on the pseudo-posterior distribution πPS(ψ|y) / πPS(ψ)LPS(ψ), have accurate frequentist coverage. Several illustrative examples are given and comparisons of πPS(ψ|y) are made to the posterior distributions based on the reference or Jeffreys priors. Some interesting conclusions emerge.