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.