Integrated likelihood methods for eliminating nuisance parameters

Elimination of nuisance parameters is a central problem in statistical infe rence and has been formally studied in virtually all approaches to inferenc e. Perhaps the least studied approach is elimination of nuisance parameters through integration, in the sense that this is viewed as an almost inciden tal byproduct of Bayesian analysis and is hence not something which is deem ed to require separate study. There is, however, considerable value in cons idering integrated likelihood on its own, especially versions arising from default or noninformative priors. In this paper, we review such common inte grated likelihoods and discuss their strengths and weaknesses relative to o ther methods.