Generalized inferential models

This paper generalizes the authors’ inferential model (IM) framework for priorfree, posterior probabilistic inference about unknown parameters. This generalization is accomplished by focusing on an association model determined by the sampling distribution of a function of the data and parameter. The advantage is that the new association model is generally easier to work with than that determined by the full sampling distribution of the data, and that the generalized IM retains the desirable frequency-calibration property of the basic IM. An important special case is when this function of data and parameters is the likelihood. Illustrative examples and further properties of this likelihood-based generalized IM are given, including extensions to handle marginal and conditional inference. The strengths of the proposed approach are showcased in two interesting marginal inference problems: the gamma mean model and a Gaussian variance components model.