A BAYESIAN APPROACH FOR ENVELOPE MODELS
A BAYESIAN APPROACH FOR ENVELOPE MODELS
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DOI：
10.1214/16-aos1449
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发表时间：
2017-02-01
影响因子：
4.5
通讯作者：
Su, Zhihua
中科院分区：
文献类型：
--
作者：
Khare, Kshitij;Pal, Subhadip;Su, Zhihua
包络模型是多元分析中估计和预测的一种新范式。使用足够的降维技术，它有可能实现显着的效率增益相比，标准模型。这个模型是由[统计学家]首先提出的。Sinica 20（2010）927-960]，并且已经适应于许多其他背景。然而，贝叶斯方法分析包络模型尚未在文献中进行了研究。在本文中，我们开发了一个全面的贝叶斯框架的估计和模型选择的包络模型的背景下，多元线性回归。我们的框架具有以下吸引人的功能。首先，我们使用矩阵宾汉分布构造一个先验的正交基矩阵的包络子空间。该先验考虑了包络模型的流形结构，并通过超参数的指定直接包含包络子空间的先验信息。该特征在更广泛的贝叶斯充分降维领域具有潜在的应用价值。其次，从所得的后验分布的采样可以通过使用具有标准关联条件的块吉布斯采样器来实现。这又促进了计算上有效的估计和模型选择。第三，与当前的频率论方法不同，我们的方法可以适应样本量小于响应数量的情况。最后，贝叶斯方法本身通过后验分布提供了全面的不确定性表征。我们说明了我们的方法在模拟和真实的数据集上的实用性。
The envelope model is a new paradigm to address estimation and prediction in multivariate analysis. Using sufficient dimension reduction techniques, it has the potential to achieve substantial efficiency gains compared to standard models. This model was first introduced by [Statist. Sinica 20 (2010) 927-960] for multivariate linear regression, and has since been adapted to many other contexts. However, a Bayesian approach for analyzing envelope models has not yet been investigated in the literature. In this paper, we develop a comprehensive Bayesian framework for estimation and model selection in envelope models in the context of multivariate linear regression. Our framework has the following attractive features. First, we use the matrix Bingham distribution to construct a prior on the orthogonal basis matrix of the envelope subspace. This prior respects the manifold structure of the envelope model, and can directly incorporate prior information about the envelope subspace through the specification of hyperparamaters. This feature has potential applications in the broader Bayesian sufficient dimension reduction area. Second, sampling from the resulting posterior distribution can be achieved by using a block Gibbs sampler with standard associated conditionals. This in turn facilitates computationally efficient estimation and model selection. Third, unlike the current frequentist approach, our approach can accommodate situations where the sample size is smaller than the number of responses. Lastly, the Bayesian approach inherently offers comprehensive uncertainty characterization through the posterior distribution. We illustrate the utility of our approach on simulated and real datasets.