Optimal prior-free probabilistic variable selection in Gaussian linear regression

Model selection is fundamental to scientific discovery, but a general framework that gives valid prior-free probabilistic assessments of the quality of individual models remains to be worked out. In this paper, we propose an inferential model (IM) approach to accomplish this for the special case of variable selection in a full-rank Gaussian linear regression model. This proposal is based on the notion of simultaneously valid inference for a collection of assertions of interest, and we develop some general results and characterizations of the corresponding optimal predictive random sets. An interesting and important by-product of our IM approach is that the resulting validity holds when the IM is projected down to a sub-model, even if that sub-model is selected by the data. In other words, the proposed IM approach automatically gives valid post-selection inference. An IM-driven variable selection procedure is proposed, and are shown to compare favorably to existing methods based on realand simulated-data examples.