Inferential Models for Linear Regression

Linear regression is arguably one of the most widely used statistical methods in applications.  However, important problems, especially variable selection, remain a challenge for classical modes of inference.  This paper develops a recently proposed framework of inferential models (IMs) in the linear regression context.  In general, an IM is able to produce meaningful probabilistic summaries of the statistical evidence for and against assertions about the unknown parameter of interest and, moreover, these summaries are shown to be properly calibrated in a frequentist sense.  Here we demonstrate, using simple examples, that the IM framework is promising for linear regression analysis --- including model checking, variable selection, and prediction --- and for uncertain inference in general.