Integrated likelihood methods for eliminating nuisance parameters

Elimination of nuisance parameters is a central problem in statistical inference and has been formally studied in virtually all approaches to inference. Perhaps the least studied approach is elimination of nuisance parameters through integration, in the sense that this is viewed as an almost incidental byproduct of Bayesian analysis and is hence not something which is deemed to require separate study. There is, however, considerable value in considering integrated likelihood on its own, especially versions arising from default or noninformative priors. In this paper, we review such common integrated likelihoods and discuss their strengths and weaknesses relative to other methods.