Summary: Consider a model parameterized by \(\mathbf{\theta}= (\psi,\mathbf{\lambda})\), where \(\psi\) is the parameter of interest. The problem of eliminating the nuisance parameter \(\mathbf{\lambda}\) can be tackled by resorting to a pseudo-likelihood function \(L^\ast(\psi)\) for \(\psi\) –namely, a function of \(\psi\) only and the data \(\mathbf y\) with properties similar to those of a likelihood function. If one treats \(L^\ast(\psi)\) as a true likelihood, the posterior distribution \(\pi^\ast(\psi|\mathbf y) \propto \pi(\psi)L^\ast(\psi)\) for \(\psi\) can be considered, where \(\pi(\psi)\) is a prior distribution on \(\psi\). 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 \(\pi^\ast(\psi|\mathbf y)\). When \(L^\ast(\psi)\) is a marginal, a conditional, or a modification of the profile likelihood, we show that \(\pi(\psi)\) 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.