Marginal Inferential Models: Prior-Free Probabilistic Inference on Interest Parameters

Abstract

The inferential models (IM) framework provides prior-free, frequency-calibrated, and 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 marginalization can be achieved, and we give conditions for marginal IM validity. We show that our approach provides exact and efficient marginal inference in several challenging problems, including a many-normal-means problem. In nonregular problems, we propose a generalized marginalization technique and prove its validity. Details are given for two benchmark examples, namely, the Behrens–Fisher and gamma mean problems.

• Belief
• Efficiency
• Nuisance parameter
• Plausibility
• Predictive random set
• Validity

Additional information

Notes on contributors

Ryan Martin

Ryan Martin, Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices, 851 S. Morgan St., Chicago, IL 60607 (E-mail: [email protected]). Chuanhai Liu, Department of Statistics, Purdue University, 250 N. University St., West Lafayette, IN 47907 (E-mail: [email protected]). The authors thank the Editor, Associate Editor, and three anonymous referees for their critical comments and suggestions, and Dr. Jing-Shiang Hwang for helpful discussion on an earlier draft of this article. This work is partially supported by the U.S. National Science Foundation, grants DMS–1007678, DMS–1208833, and DMS–1208841.

Chuanhai Liu

Ryan Martin, Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices, 851 S. Morgan St., Chicago, IL 60607 (E-mail: [email protected]). Chuanhai Liu, Department of Statistics, Purdue University, 250 N. University St., West Lafayette, IN 47907 (E-mail: [email protected]). The authors thank the Editor, Associate Editor, and three anonymous referees for their critical comments and suggestions, and Dr. Jing-Shiang Hwang for helpful discussion on an earlier draft of this article. This work is partially supported by the U.S. National Science Foundation, grants DMS–1007678, DMS–1208833, and DMS–1208841.