The Bayes oracle and asymptotic optimality of multiple testing procedures under sparsity

By Malgorzata Bogdan, A. Chakrabarti, Florian Frommlet, Jayanta Kumar Ghosh. 2010

86 citations recorded by the source corpus.

Within a Bayesian decision theoretic framework we investigate some asymptotic optimality properties of a large class of multiple testing rules. A parametric setup is considered, in which observations come from a normal scale mixture model and the total loss is assumed to be the sum of losses for individual tests. Our model can be used for testing point null hypotheses, as well as to distinguish large signals from a multitude of very small effects. A rule is defined to be asymptotically Bayes optimal under sparsity (ABOS), if within our chosen asymptotic framework the ratio of its Bayes risk and that of the Bayes oracle (a rule which minimizes the Bayes risk) converges to one. Our main interest is in the asymptotic scheme where the proportion p of "true" alternatives converges to zero. We fully characterize the class of fixed threshold multiple testing rules which are ABOS, and hence derive conditions for the asymptotic optimality of rules controlling the Bayesian False Discovery Rate (BFDR). We finally provide conditions under which the popular Benjamini-Hochberg (BH) and Bonferroni procedures are ABOS and show that for a wide class of sparsity levels, the threshold of the former can be approximated by a nonrandom threshold.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1775719a073b2f510ba5454a69f04aa6051fbff0

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Citations in the corpus

SLOPE—ADAPTIVE VARIABLE SELECTION VIA CONVEX OPTIMIZATION (2014, background): “This lemma, whose proof is in the supplementary Appendix [Bogdan et al. (2015)], guarantees that the FastProxSL1 algorithm finds the solution to (2.”
Asymptotic Properties of Bayes Risk of a General Class of Shrinkage Priors in Multiple Hypothesis Testing Under Sparsity (2013, methodology): “(2009), Scott and Berger (2010), Bogdan et al. (2008) and Datta and Ghosh (2013). We refer to van der Pas et al.”
Adaptive posterior contraction rates for the horseshoe (2017, background): “Thus, the ‘extremely sparse’ case, as identified by [5], where pn → s ∈ (0,∞] and log(pn)/ log(n) → 0, is not entirely covered by Condition 3.”
Distribution-free Multiple Testing (2016, methodology): “Most other publications that discuss oracle bounds, such as [5, 16, 24, 28, 29], operate in a setting where the statistics have the same null distribution and the same alternative distribution, and consider an oracle that provides these two distributions together with the number of false null hypoth”