Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences
Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences
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DOI：
10.1214/009053604000000030
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发表时间：
2004-08-01
影响因子：
4.5
通讯作者：
Silverman, BW
中科院分区：
文献类型：
--
作者：
Johnstone, IM;Silverman, BW
提出并研究了一种经验贝叶斯方法，用于估计高斯白色噪声中可能观测到的稀疏序列。考虑的先验是概率为零的原子和重尾密度γ的混合物，混合权重由边际最大似然法选择，希望在稀疏序列和密集序列之间进行调整。如果使用后验中值进行估计，则这是随机阈值处理过程。也可以使用采用相同阈值的其他阈值规则。由边际最大似然法选择的阈值上的概率界限导致长度为n的信号序列的类的总体风险界限，允许各种和程度的稀疏性。所考虑的信号类别是“近黑”序列，其中仅允许比例eta为非零，以及具有由eta限定的归一化l（p）范数的序列，其中eta > 0且0 < p小于或等于2。估计误差通过平均第q个功率损耗来测量，0 < q小于或等于2。对于所考虑的所有类别，并且对于（0，2）中的所有q，该方法在各种速率下实现最佳估计速率为n ->无穷大和eta -> 0，并且在这个意义上自动适应于基础信号的稀疏性或其他方面。此外，风险是一致有界的所有信号。如果使用后验均值作为估计量，结果仍然适用于q > 1。仿真结果显示了良好的性能。对于适当选择的函数γ，该方法在计算上易于处理，并且软件可用。扩展到一个修改的阈值相关的非常稀疏的序列的估计方法也被认为是。
An empirical Bayes approach to the estimation of possibly sparse sequences observed in Gaussian white noise is set out and investigated. The prior considered is a mixture of an atom of probability at zero and a heavy-tailed density gamma, with the mixing weight chosen by marginal maximum likelihood, in the hope of adapting between sparse and dense sequences. If estimation is then carried Out using the posterior median, this is a random thresholding procedure. Other thresholding rules employing the same threshold can also be used. Probability bounds on the threshold chosen by the marginal maximum likelihood approach lead to overall risk bounds over classes of signal sequences of length n, allowing for sparsity of various kinds and degrees. The signal classes considered are "nearly black" sequences where only a proportion eta is allowed to be nonzero, and sequences with normalized l(p) norm bounded by eta, for eta > 0 and 0 < p less than or equal to 2. Estimation error is measured by mean qth power loss, for 0 < q less than or equal to 2. For all the classes considered, and for all q in (0, 2), the method achieves the optimal estimation rate as n --> infinity and eta --> 0 at various rates, and in this sense adapts automatically to the sparseness or otherwise of the underlying signal. In addition the risk is uniformly bounded over all signals. If the posterior mean is used as the estimator, the results still hold for q > 1. Simulations show excellent performance. For appropriately chosen functions gamma, the method is computationally tractable and software is available. The extension to a modified thresholding method relevant to the estimation of very sparse sequences is also considered.