Asymptotic Properties of Bayes Risk of a General Class of Shrinkage Priors in Multiple Hypothesis Testing Under Sparsity
Asymptotic Properties of Bayes Risk of a General Class of Shrinkage Priors in Multiple Hypothesis Testing Under Sparsity
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稀疏性下多重假设检验中一般类收缩先验的贝叶斯风险的渐近性质
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发表时间：
2013
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通讯作者：
A. Chakrabarti
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作者：
P. Ghosh;Xueying Tang;M. Ghosh;A. Chakrabarti
考虑独立正态观测值均值的同时检验问题。本文在贝叶斯决策理论框架下，研究了一类单组收缩先验引起的若干检验规则的渐近最优性，其中总损失为错误分类假设的数量。我们假设数据为两组正态混合模型，并考虑Bogdan等人（2011）采用的渐近框架，他们在多次检验的背景下引入了稀疏性下渐近贝叶斯最优性的概念。所研究的单组先验的一般类别足够丰富，其中包括三个参数的β族，广义双帕累托先验，特别是马蹄先验，正态指数先验和Strawderman-Berger先验。我们建立了在我们所选择的渐近框架内，所研究的多个测试规则渐近地达到Bayes Oracle风险的乘因子，并且风险中的常数接近Oracle风险中的常数。这与Datta和Ghosh（2013）对Carvalho等人（2009、2010）引入的基于马蹄估计量的多重检验规则的结果相似。我们进一步表明，在对潜在稀疏度参数的非常温和的假设下，基于van der Pas等人（2014）提出的相应全局收缩参数的经验贝叶斯估计的诱导决策规则，可以渐近地获得相同乘因子的最优贝叶斯风险。我们提供了一个适用于所研究的一般先验类别的统一论证。在此过程中，我们解决了Datta和Ghosh（2013）提出的关于广义双Pareto先验的最优性的猜想。我们的工作还表明，Datta和Ghosh（2013）的结果可以进一步改进。
Consider the problem of simultaneous testing for the means of independent normal observations. In this paper, we study some asymptotic optimality properties of certain multiple testing rules induced by a general class of one-group shrinkage priors in a Bayesian decision theoretic framework, where the overall loss is taken as the number of misclassified hypotheses. We assume a two-groups normal mixture model for the data and consider the asymptotic framework adopted in Bogdan et al. (2011) who introduced the notion of asymptotic Bayes optimality under sparsity in the context of multiple testing. The general class of one-group priors under study is rich enough to include, among others, the families of three parameter beta, generalized double Pareto priors, and in particular the horseshoe, the normal-exponential-gamma and the Strawderman-Berger priors. We establish that within our chosen asymptotic framework, the multiple testing rules under study asymptotically attain the risk of the Bayes Oracle up to a multiplicative factor, with the constant in the risk close to the constant in the Oracle risk. This is similar to a result obtained in Datta and Ghosh (2013) for the multiple testing rule based on the horseshoe estimator introduced in Carvalho et al. (2009, 2010). We further show that under very mild assumption on the underlying sparsity parameter, the induced decision rules based on an empirical Bayes estimate of the corresponding global shrinkage parameter proposed by van der Pas et al. (2014), attain the optimal Bayes risk up to the same multiplicative factor asymptotically. We provide a unifying argument applicable for the general class of priors under study. In the process, we settle a conjecture regarding optimality property of the generalized double Pareto priors made in Datta and Ghosh (2013). Our work also shows that the result in Datta and Ghosh (2013) can be improved further.