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Inferential Models: A Framework for Prior-Free Posterior Probabilistic Inference

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  • Ryan Martin
  • Chuanhai Liu

Abstract

Posterior probabilistic statistical inference without priors is an important but so far elusive goal. Fisher's fiducial inference, Dempster--Shafer theory of belief functions, and Bayesian inference with default priors are attempts to achieve this goal but, to date, none has given a completely satisfactory picture. This article presents a new framework for probabilistic inference, based on inferential models (IMs), which not only provides data-dependent probabilistic measures of uncertainty about the unknown parameter, but also does so with an automatic long-run frequency-calibration property. The key to this new approach is the identification of an unobservable auxiliary variable associated with observable data and unknown parameter, and the prediction of this auxiliary variable with a random set before conditioning on data. Here we present a three-step IM construction, and prove a frequency-calibration property of the IM's belief function under mild conditions. A corresponding optimality theory is developed, which helps to resolve the nonuniqueness issue. Several examples are presented to illustrate this new approach.

Suggested Citation

  • Ryan Martin & Chuanhai Liu, 2013. "Inferential Models: A Framework for Prior-Free Posterior Probabilistic Inference," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(501), pages 301-313, March.
  • Handle: RePEc:taf:jnlasa:v:108:y:2013:i:501:p:301-313
    DOI: 10.1080/01621459.2012.747960
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    References listed on IDEAS

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    1. Xie, Minge & Singh, Kesar & Strawderman, William E., 2011. "Confidence Distributions and a Unifying Framework for Meta-Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 106(493), pages 320-333.
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    Cited by:

    1. Xuhua Liu & Xingzhong Xu, 2016. "Confidence distribution inferences in one-way random effects model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 59-74, March.
    2. Jin, Hua & Li, Song & Jin, Yaolan, 2016. "The IM-based method for testing the non-inferiority of odds ratio in matched-pairs design," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 145-151.
    3. Piero Veronese & Eugenio Melilli, 2015. "Fiducial and Confidence Distributions for Real Exponential Families," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(2), pages 471-484, June.
    4. Xuhua Liu & Xingzhong Xu, 2016. "Confidence distribution inferences in one-way random effects model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 59-74, March.
    5. Dungang Liu & Regina Y. Liu & Minge Xie, 2015. "Multivariate Meta-Analysis of Heterogeneous Studies Using Only Summary Statistics: Efficiency and Robustness," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 326-340, March.
    6. Randy C. S. Lai & Jan Hannig & Thomas C. M. Lee, 2015. "Generalized Fiducial Inference for Ultrahigh-Dimensional Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 760-772, June.
    7. Michael P. Fay & Michael A. Proschan & Erica Brittain, 2015. "Combining one-sample confidence procedures for inference in the two-sample case," Biometrics, The International Biometric Society, vol. 71(1), pages 146-156, March.

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