Prior distributions from pseudo-likelihoods in the presence of nuisance parameters

Consider a model parameterized by q = (p, l), where p is the parameter of interest. The problem of eliminating the nuisance parameter l can be tackled by resorting to a pseudo-likelihood function L*(p) for p — namely, a function of p only and the data y with properties similar to those of a likelihood function. If one treats L*(p) as a true likelihood, the posterior distribution p*(p | y) } p(p)L*(p) for p can be considered, where p(p) is a prior distribution on p. The goal of this article is to construct probability matching priors for a scalar parameterof interest only (i.e., priors for which Bayesian and frequentist inference agree to some order of approximation) to be used in p*(p | y). When L*(p) is a marginal, a conditional, or a modification of the profile likelihood, we show that p(p) 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.