Marginal inferential models

Inferential models (IMs) provide a general framework for prior-free, frequencycalibrated, posterior probabilistic inference. The fundamental idea is the use of unobservable auxiliary variables to describe the underlying uncertainty about the parameter of interest. When nuisance parameters are present, a marginalization step can reduce the dimension of the auxiliary variable, which in turn leads to more efficient inference. Unlike classical marginalization procedures that use optimization or integration, the proposed framework works with a set union operation. Sufficient conditions are given for when this marginalization operation can be performed without loss of information, and in such cases we prove that an appropriately constructed IM is calibrated, in a frequentist sense, for marginal inference. In problems where these sufficient conditions are not met, we propose a generalized marginalization technique based on parameter expansion that leads to possibly conservative marginal inference. Details are given for two benchmark examples: Stein’s many-normal-means problem and the Behrens–Fisher problem.