Marginal inferential models: prior-free probabilistic inference on interest parameters

Inferential models (IMs) provide a general framework for 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 and efficient marginalization can be achieved, and we give conditions for marginal IM validity. We show that our approach provides efficient marginal inference in several challenging problems, including a many-normal-means problem, and does not suffer from common marginalization paradoxes. In non-regular problems, we propose a generalized marginalization technique which is valid and also paradox-free. Details are given for two benchmark examples, namely, the Behrens--Fisher and gamma mean problems.