We study asymptotic properties of Bayesian multiple testing procedures and provide sufficient conditions for strong consistency under general dependence structure. We also consider a novel Bayesian multiple testing procedure and associated error measures that coherently accounts for the dependence structure present in the model. We advocate posterior versions of FDR and FNR as appropriate error rates and show that their asymptotic convergence rates are directly associated with the Kullback-Leibler divergence from the true model. Our results hold even when the class of postulated models is misspecified. We illustrate our results in a variable selection problem with autoregressive response variables, and compare the new Bayesian procedure with some existing methods through extensive simulation studies in the variable selection problem. Superior performance of the new procedure compared to the others vindicate that proper exploitation of the dependence structure by multiple testing methods is indeed important. Moreover, we obtain encouraging results in a real, maize data context, where we select influential marker variables.

中文翻译：

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可能模型错误指定下的相关贝叶斯多重测试程序的渐近理论

我们研究了贝叶斯多重测试程序的渐近特性，并为一般依赖结构下的强一致性提供了充分条件。我们还考虑了一种新颖的贝叶斯多重测试程序和相关的误差度量，它们一致地解释了模型中存在的依赖结构。我们主张将 FDR 和 FNR 的后验版本作为适当的错误率，并表明它们的渐近收敛率与真实模型的 Kullback-Leibler 散度直接相关。即使错误指定了假设模型的类别，我们的结果也成立。我们在具有自回归响应变量的变量选择问题中说明了我们的结果，并通过对变量选择问题的广泛模拟研究将新的贝叶斯程序与一些现有方法进行了比较。与其他程序相比，新程序的优越性能证明通过多种测试方法正确利用依赖结构确实很重要。此外，我们在真实的玉米数据环境中获得了令人鼓舞的结果，我们选择了有影响力的标记变量。