Statistica Sinica 9(1999), 713-724

ON THE RELATIONSHIP BETWEEN BAYESIAN AND

NON-BAYESIAN ELIMINATION OF NUISANCE

PARAMETERS

Thomas A. Severini

Northwestern University

Abstract: Consider a statistical model parameterized by a scalar parameter of interest and a nuisance parameter . Many methods of inference are based on a pseudo-likelihood'' function, a function of the data and that has properties similar to those of a likelihood function. Commonly used pseudo-likelihood functions include conditional likelihood functions, marginal likelihood functions, and profile likelihood functions. From the Bayesian point of view, elimination of is easily achieved by integrating the likelihood function with respect to a conditional prior density ; this approach has some well-known optimality properties. In this paper, we study how close certain pseudo-likelihood functions are to being of Bayesian form. It is shown that many commonly used non-Bayesian methods of eliminating correspond to Bayesian elimination of to a high degree of approximation.

Key words and phrases: Conditional likelihood, integrated likelihood, marginal likelihood, profile likelihood.