Mixtures, envelopes and hierarchical duality
Mixtures, envelopes and hierarchical duality
复制标题
混合、包络和层次二元性
DOI：
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复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
James G. Scott
中科院分区：
文献类型：
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作者：
Nicholas G. Polson;James G. Scott
我们开发的混合物和包络表示的目标函数，经常出现在统计之间的连接。我们用“层次二重性”这个术语来指这种联系。我们的研究结果表明，一个有趣的和以前未充分利用的边缘化和配置之间的关系，或相当于之间的Fenchel-Moreau定理凸函数和Bernstein-Widder定理的拉普拉斯变换。我们给出了几组不同的条件下，这样的对偶结果获得。然后，我们扩展现有的工作在几个方面，包括新的概括方差均值模型和多元高斯位置模型的包络表示。这为近似梯度法提供了一种优雅的缺失数据解释，这是一种广泛使用的机器学习算法。我们展示了几个统计应用，其中提出的框架导致易于实现的算法，包括融合套索的鲁棒版本，通过趋势过滤的非线性分位数回归和二项式融合双帕累托模型。示例代码可在GitHub上获得，网址为https://github.com/jgscott/hierduals。
We develop a connection between mixture and envelope representations of objective functions that arise frequently in statistics. We refer to this connection by using the term ‘hierarchical duality’. Our results suggest an interesting and previously underexploited relationship between marginalization and profiling, or equivalently between the Fenchel–Moreau theorem for convex functions and the Bernstein–Widder theorem for Laplace transforms. We give several different sets of conditions under which such a duality result obtains. We then extend existing work on envelope representations in several ways, including novel generalizations to variance–mean models and to multivariate Gaussian location models. This turns out to provide an elegant missing data interpretation of the proximal gradient method, which is a widely used algorithm in machine learning. We show several statistical applications in which the framework proposed leads to easily implemented algorithms, including a robust version of the fused lasso, non‐linear quantile regression via trend filtering and the binomial fused double‐Pareto model. Code for the examples is available on GitHub at https://github.com/jgscott/hierduals.
DOI：
10.1214/10-sts336
发表时间：
2010-08-01
期刊：
Statistical science : a review journal of the Institute of Mathematical Statistics
影响因子：
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作者：
Zhou H;Lange K;Suchard MA
通讯作者：
Suchard MA
影响因子：
4.4
作者：
Gramacy, Robert B.;Polson, Nicholas G.
通讯作者：
Polson, Nicholas G.