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.