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On the prevalence of information inconsistency in normal linear models

Author

Listed:
  • Joris Mulder

    (Tilburg University
    Jheronimus Academy of Data Science)

  • James O. Berger

    (Duke University)

  • Víctor Peña

    (Baruch College Zicklin School of Business)

  • M. J. Bayarri

    (Universitat de Valéncia)

Abstract

Informally, ‘information inconsistency’ is the property that has been observed in some Bayesian hypothesis testing and model selection scenarios whereby the Bayesian conclusion does not become definitive when the data seem to become definitive. An example is that, when performing a t test using standard conjugate priors, the Bayes factor of the alternative hypothesis to the null hypothesis remains bounded as the t statistic grows to infinity. The goal of this paper is to thoroughly investigate information inconsistency in various Bayesian testing problems. We consider precise hypothesis tests, one-sided hypothesis tests, and multiple hypothesis tests under normal linear models with dependent observations. Standard priors are considered, such as conjugate and semi-conjugate priors, as well as variations of Zellner’s g prior (e.g., fixed g priors, mixtures of g priors, and adaptive (data-based) g priors). It is shown that information inconsistency is a widespread problem using standard priors while certain theoretically recommended priors, including scale mixtures of conjugate priors and adaptive priors, are information consistent.

Suggested Citation

  • Joris Mulder & James O. Berger & Víctor Peña & M. J. Bayarri, 2021. "On the prevalence of information inconsistency in normal linear models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 103-132, March.
    • Handle: RePEc:spr:testjl:v:30:y:2021:i:1:d:10.1007_s11749-020-00704-4
    DOI: 10.1007/s11749-020-00704-4
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    References listed on IDEAS

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    1. Klugkist, Irene & Hoijtink, Herbert, 2007. "The Bayes factor for inequality and about equality constrained models," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6367-6379, August.
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    3. Liang, Feng & Paulo, Rui & Molina, German & Clyde, Merlise A. & Berger, Jim O., 2008. "Mixtures of g Priors for Bayesian Variable Selection," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 410-423, March.
    4. Hansen M. H & Yu B., 2001. "Model Selection and the Principle of Minimum Description Length," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 746-774, June.
    5. Sellke T. & Bayarri M. J. & Berger J. O., 2001. "Calibration of rho Values for Testing Precise Null Hypotheses," The American Statistician, American Statistical Association, vol. 55, pages 62-71, February.
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