Abstract: Consider a statistical model parameterized by a scalar parameter of interestand 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.