For high-dimensional hierarchical models, consider exchangeability of effects across covariates instead of across datasets
For high-dimensional hierarchical models, consider exchangeability of effects across covariates instead of across datasets
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发表时间：
2021-07
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通讯作者：
Brian L. Trippe;H. Finucane;Tamara Broderick
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作者：
Brian L. Trippe;H. Finucane;Tamara Broderick
分层贝叶斯方法可以在多个相关的回归问题中共享信息。虽然标准做法是将回归参数（效应）建模为（1）在数据集之间可交换，（2）在协变量之间具有不同程度的相关性，但我们发现，当协变量的数量超过数据集的数量时，这种方法的统计性能较差。例如，在统计遗传学中，我们可能会对成千上万个个体（反应）的数十个特征（定义数据集）进行回归，以获得数百万个遗传变异（协变量）。当分析师的协变量多于数据集时，我们认为将效应建模为（1）协变量之间的可交换性和（2）数据集之间不同程度的相关性通常更自然。为此，我们提出了一个层次模型表达我们的替代观点。我们设计了一个经验贝叶斯估计学习数据集之间的相关程度。我们开发的理论表明，我们的方法优于经典的方法时，协变量的数量占主导地位的数据集的数量，并证实了这一结果经验上的几个高维多元回归和分类问题。
Hierarchical Bayesian methods enable information sharing across multiple related regression problems. While standard practice is to model regression parameters (effects) as (1) exchangeable across datasets and (2) correlated to differing degrees across covariates, we show that this approach exhibits poor statistical performance when the number of covariates exceeds the number of datasets. For instance, in statistical genetics, we might regress dozens of traits (defining datasets) for thousands of individuals (responses) on up to millions of genetic variants (covariates). When an analyst has more covariates than datasets, we argue that it is often more natural to instead model effects as (1) exchangeable across covariates and (2) correlated to differing degrees across datasets. To this end, we propose a hierarchical model expressing our alternative perspective. We devise an empirical Bayes estimator for learning the degree of correlation between datasets. We develop theory that demonstrates that our method outperforms the classic approach when the number of covariates dominates the number of datasets, and corroborate this result empirically on several high-dimensional multiple regression and classification problems.