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

Consider a statistical model parameterized by a scalar parameter of interes t a and theta nuisance parameter lambda. Many methods of inference are base d on a "pseudo-likelihood'' function, a function of the data and theta that has properties similar to those of a likelihood function. Commonly used ps eudo-likelihood functions include conditional likelihood functions, margina l 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\ theta); this approach has some well-known optimality properties. In this pa per, 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.