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Abstract:
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We first provide results from a simulation based evaluation of the frequentist characteristics of the Bayesian approach to multiple testing. We then turn to the issue of deciding when to reject a null. In the absence of a specified loss function, we suggest the use of frequentist error rates to determine a suitable cut-off for the posterior probability. We then introduce a loss function that may be useful when sparsity is deemed desirable. This sparsity inducing loss function (SIL) can be thought of as a modified version of the usual 0-1 loss. It assigns a smaller loss for non-discovery when the signal is close to zero; it is bounded, and is computationally similar. We present a numerical comparison of the performance of SIL with the 0-1 loss and with some frequentist approaches, using simulated and real data. Additionally, an empirical Bayes approach is used with SIL and compared with the procedures given in Malgorzata et al. (2007), and the Benjamini Hochberg method, using simulation.
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