On the relationship between Bayesian and non-Bayesian elimination of nuisance parameters

Consider a statistical model parameterized by a scalar parameter of in- terest θ and a nuisance parameter λ. Many methods of inference are based on a function, a function of the data and θ that has properties similar to those of a likelihood function. Commonly used pseudo-likelihood func- tions include conditional likelihood functions, marginal likelihood functions, and profile likelihood functions. From the Bayesian point of view, elimination of λ is easily achieved by integrating the likelihood function with respect to a conditional prior density π(λ|θ); 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 λ correspond to Bayesian elimination of λ to a high degree of approxi-