Implementing matching priors for frequentist inference

Nuisance parameters do not pose any problems in Bayesian inference as marginalisation allows for study of the posterior distribution solely in terms of the parameter of interest. However, no general solution is available for removing nuisance parameters under the frequentist paradigm. In this paper, we merge the two approaches to construct a general procedure for frequentist elimination of nuisance parameters through the use of matching priors. In particular, we perform Bayesian marginalisation with respect to a prior distribution under which posterior inferences have approximate frequentist validity. Matching priors are constructed as solutions to a partial differential equation. Unfortunately, except in simple cases, these partial differential equations do not yield to analytical nor even standard numerical methods of solution. We present a numerical/Monte Carlo algorithm for obtaining the matching prior, in general, as a solution to the appropriate partial differential equation and draw posterior inferences. To be specific, we develop an automated routine through an implementation of the Metropolis-Hastings algorithm for deriving frequentist valid inferences via the matching prior. We illustrate our results in the contexts of fitting random effects models, fitting logistic regression models and fitting teratological data by beta-binomial models.