Sankhya: The Indian Journal of Statistics

2006, Volume 68, Pt. 3, 436--460

Bayesian Maximum a posteriori Multiple Testing Procedure

Felix Abramovich, Tel Aviv University, Tel Aviv, Israel
Claudia Angelini, Consiglio Nazionale delle Ricerche, Napoli, Italy

SUMMARY. We consider a Bayesian approach to multiple hypothesis testing. A hierarchical prior model is based on imposing a prior distribution $\pi(k)$ on the number of hypotheses arising from alternatives (false nulls). We then apply the maximum a posteriori (MAP) rule to find the most likely configuration of null and alternative hypotheses. The resulting MAP procedure and its closely related step-up and step-down versions compare ordered Bayes factors of individual hypotheses with a sequence of critical values depending on the prior. We discuss the relations between the proposed MAP procedure and the existing frequentist and Bayesian counterparts. A more detailed analysis is given for the normal data, where we show, in particular, that by choosing a specific $\pi(k)$, the MAP procedure can mimic several known familywise error (FWE) and false discovery rate (FDR) controlling procedures. The performance of MAP procedures is illustrated on a simulated example.

AMS (2000) subject classification. Primary 62F15, 62F03.

Key words and phrases. Bayes factor, false discovery rate, familywise error, hierarchical prior, maximum a posteriori rule, multiple hypothesis testing, $p$-value.

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