On the role of the prior in multiplicity adjustment

Journal of Statistical Theory and Practice - Tập 10 - Trang 263-290 - 2016
Dandan Li1, Siva Sivaganesan1
1Department of Mathematical Sciences, University of Cincinnati, Cincinnati, USA

Tóm tắt

Multiplicity adjustment in Bayesian analysis is achieved through the use of a prior distribution, for the probability that a variable is in the (unknown) model in the context of model selection, or for the probability that a null hypothesis is true in the context of multiple testing. However, it is not obvious how the prior distribution brings about multiplicity adjustment. In 2010 Scott and Berger stated there is an “air of paradox” in how multiplicity adjustment is achieved in the fully Bayesian approach. They gave useful insight into the role of the prior distribution in multiplicity adjustment by using the prior odds ratio (POR), the ratio of prior probabilities of a smaller model to a larger model, and used a uniform distribution for the prior in their illustration. In this article, we identify certain characteristics of POR based on the uniform prior that help explain the role of the prior in multiplicity adjustment, and provide generalizations of these properties to more general priors. We use these results to develop a summary measure to quantify the degree of multiplicity adjustment inherent in a prior distribution, and discuss the relative roles of the hyperparameters in a Beta prior or mixtures of them. We use simulation results to illustrate these findings.

Tài liệu tham khảo

Bayarri, M. J., J. O. Berger, A. Forte, and G. García-Donato. 2012. Criteria for Bayesian model choice with application to variable selection. Annals of Statistics 40(3):1550–77. Bogdan, M., J. Ghosh, and R. Doerge. 2004. Modifying the Schwarz Bayesian information criterion to locate multiple interacting quantitative trait loci. Genetics 167(2):989–99. Bogdan, M., A. Chakrabarti, F. Frommlet, and J. K. Ghosh. 2011. Asymptotic Bayes-optimality under sparsity of some multiple testing procedures. Annals of Statistics 39(3):1551–79. Clyde, M. A., J. Ghosh, and M. Littman. 2011. Bayesian adaptive sampling for variable selection and model averaging. Journal of Computational and Graphical Statistics 20:80–101. Ghosal, S. 2001. Convergence rates for density estimation with Bernstein polynomials. Annals of Statistics 29(1):1264–80. Liang, F., R. Paulo, G. Molina, M. A. Clyde, and J. O. Berger. 2008. Mixtures of g priors for Bayesian variable selection. Journal of the American Statistical Association 103(5):410–23. Maruyama, Y., and E. I. George. 2011. Fully Bayes factors with a generalized g-prior. Annals of Statistics 39(481):2740–65. Pétrone, S., and L. Wasserman. 2002. Consistency of Bernstein polynomial posteriors. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 64(1):79–100. Scott, J. G., and J. O. Berger. 2006. An exploration of aspects of Bayesian multiple testing. Journal of Statistical Planning and Inference 136(7):2144–62. Scott, J. G., and J. O. Berger. 2010. Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem. Annals of Statistics 38(5):2587–619. Zellner, A., and A. Siow. 1980. Posterior odds ratios for selected regression hypotheses. In Bayesian Statistics: Proceedings of the First International Meeting held in Valencia (Spain), Vol. 1, ed. J. M. Bernardo, M. H. DeGroot, D. V. Lindley, and A. F. M. Smith, 585–603. Valencia, Spain: University Press.