  The inferential models (IM) framework provides prior-free,
frequency-calibrated, posterior probabilistic inference. The key is the use of
random sets to predict unobservable auxiliary variables connected to the
observable data and unknown parameters. When nuisance parameters are present, a
marginalization step can reduce the dimension of the auxiliary variable which,
in turn, leads to more efficient inference. For regular problems, exact
marginalization can be achieved, and we give conditions for marginal IM
validity. We show that our approach provides exact and efficient marginal
inference in several challenging problems, including a many-normal-means
problem. In non-regular problems, we propose a generalized marginalization
technique and prove its validity. Details are given for two benchmark examples,
namely, the Behrens--Fisher and gamma mean problems.
