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

Consider a model parameterized by u 1⁄4 (c, l), where c is the parameter of interest. The problem of eliminating the nuisance parameter l can be tackled by resorting to a pseudo-likelihood function L*(c) for c—namely, a function of c only and the data y with properties similar to those of a likelihood function. If one treats L*(c) as a true likelihood, the posterior distribution p*(c | y) } p(c)L*(c) for c can be considered, where p(c) is a prior distribution on c. 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 p*(c | y). When L*(c) is a marginal, a conditional, or a modification of the profile likelihood, we show that p(c) 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.