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 $\theta$ and a nuisance parameter $\lambda$. Many methods of inference are based on a ``pseudo-likelihood'' function, a function of the data and $\theta$ 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 $\lambda$ is easily achieved by integrating the likelihood function with respect to a conditional prior density ${ \pi(\lambda\vert \theta)}$; 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 $\lambda$ correspond to Bayesian elimination of $\lambda$ to a high degree of approximation.

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