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On the use of non‐local prior densities in Bayesian hypothesis tests

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  • Valen E. Johnson
  • David Rossell

Abstract

Summary. We examine philosophical problems and sampling deficiencies that are associated with current Bayesian hypothesis testing methodology, paying particular attention to objective Bayes methodology. Because the prior densities that are used to define alternative hypotheses in many Bayesian tests assign non‐negligible probability to regions of the parameter space that are consistent with null hypotheses, resulting tests provide exponential accumulation of evidence in favour of true alternative hypotheses, but only sublinear accumulation of evidence in favour of true null hypotheses. Thus, it is often impossible for such tests to provide strong evidence in favour of a true null hypothesis, even when moderately large sample sizes have been obtained. We review asymptotic convergence rates of Bayes factors in testing precise null hypotheses and propose two new classes of prior densities that ameliorate the imbalance in convergence rates that is inherited by most Bayesian tests. Using members of these classes, we obtain analytic expressions for Bayes factors in linear models and derive approximations to Bayes factors in large sample settings.

Suggested Citation

  • Valen E. Johnson & David Rossell, 2010. "On the use of non‐local prior densities in Bayesian hypothesis tests," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(2), pages 143-170, March.
  • Handle: RePEc:bla:jorssb:v:72:y:2010:i:2:p:143-170
    DOI: 10.1111/j.1467-9868.2009.00730.x
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    File URL: https://doi.org/10.1111/j.1467-9868.2009.00730.x
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    Citations

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    Cited by:

    1. T S Shively & S G Walker, 2018. "On Bayes factors for the linear model," Biometrika, Biometrika Trust, vol. 105(3), pages 739-744.
    2. Ho-Hsiang Wu & Marco A. R. Ferreira & Mohamed Elkhouly & Tieming Ji, 2020. "Hyper Nonlocal Priors for Variable Selection in Generalized Linear Models," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(1), pages 147-185, February.
    3. Shi, Guiling & Lim, Chae Young & Maiti, Tapabrata, 2019. "Model selection using mass-nonlocal prior," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 36-44.
    4. D. Fouskakis, 2019. "Priors via imaginary training samples of sufficient statistics for objective Bayesian hypothesis testing," METRON, Springer;Sapienza Università di Roma, vol. 77(3), pages 179-199, December.
    5. Li, Hanning & Pati, Debdeep, 2017. "Variable selection using shrinkage priors," Computational Statistics & Data Analysis, Elsevier, vol. 107(C), pages 107-119.
    6. Fetene B. Tekle & Dereje W. Gudicha & Jeroen K. Vermunt, 2016. "Power analysis for the bootstrap likelihood ratio test for the number of classes in latent class models," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 10(2), pages 209-224, June.
    7. Mark F. J. Steel, 2020. "Model Averaging and Its Use in Economics," Journal of Economic Literature, American Economic Association, vol. 58(3), pages 644-719, September.
    8. David Rossell & Donatello Telesca, 2017. "Nonlocal Priors for High-Dimensional Estimation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(517), pages 254-265, January.
    9. Gelman Andrew & Robert Christian P. & Rousseau Judith, 2013. "Inherent difficulties of non-Bayesian likelihood-based inference, as revealed by an examination of a recent book by Aitkin," Statistics & Risk Modeling, De Gruyter, vol. 30(2), pages 105-120, June.
    10. Scott D. Goddard & Valen E. Johnson, 2016. "Restricted most powerful Bayesian tests for linear models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(4), pages 1162-1177, December.
    11. Christian P. Robert, 2013. "On the jeffreys-Lindley's Paradox," Working Papers 2013-46, Center for Research in Economics and Statistics.
    12. Davide Altomare & Guido Consonni & Luca La Rocca, 2013. "Objective Bayesian Search of Gaussian Directed Acyclic Graphical Models for Ordered Variables with Non-Local Priors," Biometrics, The International Biometric Society, vol. 69(2), pages 478-487, June.
    13. Nilotpal Sanyal & Marco A. R. Ferreira, 2017. "Bayesian Wavelet Analysis Using Nonlocal Priors with an Application to fMRI Analysis," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(2), pages 361-388, November.
    14. David G Jenkins & Pedro F Quintana-Ascencio, 2020. "A solution to minimum sample size for regressions," PLOS ONE, Public Library of Science, vol. 15(2), pages 1-15, February.
    15. Shi, Guiling & Lim, Chae Young & Maiti, Tapabrata, 2019. "Bayesian model selection for generalized linear models using non-local priors," Computational Statistics & Data Analysis, Elsevier, vol. 133(C), pages 285-296.
    16. Guido Consonni & Luca La Rocca, 2010. "Moment Priors for Bayesian Model Choice with Applications to Directed Acyclic Graphs," Quaderni di Dipartimento 115, University of Pavia, Department of Economics and Quantitative Methods.
    17. Christine Peterson & Francesco C. Stingo & Marina Vannucci, 2015. "Bayesian Inference of Multiple Gaussian Graphical Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 159-174, March.

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