A Theoretical Foundation for Applications of Bayesian Variable Selection

贝叶斯变量选择应用的理论基础

• 批准号: 0706885
• 负责人: Wenxin Jiang
• 金额: $ 3.81万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706885&HistoricalAwards=false
• 关键词: Theoretical; Foundation; Applications; Bayesian; Variable

In the popular approach of `Bayesian Variable Selection' (BVS), one uses prior and posterior distributions to select a subset of candidate variables to enter the model. In some examples, the number of candidate variables `K' can be much larger than the number of study units `n'. This idea has been applied to various statistical models, e.g., regression, graphical models, survival analysis, and cluster analysis. Despite its popularity, theoretical properties, especially frequentist convergence properties, have not been well established. Recently, the investigators have successfully studied the frequentist convergence properties (consistency, convergence rates, and predictive performances) of BVS for generalized linear models. A completely new direction is considered in this project to study BVS with a Gibbs posterior originating in statistical mechanics. In contrast to the usual Bayesian prior which is constructed from a likelihood function, the Gibbs posterior is constructed from a risk function of practical interest (such as the classification error) and aims at minimizing a risk function without modeling the data probablistically. This can improve the performance over the usual Bayesian approach, since the usual Bayesian approach depends on a probability model which may be misspecified. The investigators studies the statistical performance of BVS with a Gibbs posterior constructed for the purpose of classification. Conditions are provided so that BVS will achieve good classification performance, even in the presence of high dimensionality (Kn). BVS has multi-disciplinary applications that include various practices of data mining, where a few important variables are to be selected from many candidates and used for prediction and decision making, e.g., pattern recognition, fraud detection, homeland security, customer-oriented marketing decisions, machine learning, microarray analysis, and bioinformatics. The applications typicially involve many candidate variables (sometimes much more than the sample size). BVS, through selecting a few important variables, can be very helpful for interpretation, prediction, and decision making in each of these applications, despite the potentially high dimensionality. The current project will provide a theoretical framework and conditions under which BVS with a Gibbs posterior will be nearly optimal in some sense, despite high dimensionality, therefore providing theoretical justification for this important technique. A solid theoretical foundation for BVS will also lead to better interpretations of the results obtained from BVS, and provide useful information for practitioners on specification of the prior distribution. Such a good theoretical understanding will likely lead to improvement of empirical performance under many circumstances.

在“贝叶斯变量选择”（BVS）的流行方法中，人们使用先验和后验分布来选择候选变量的子集进入模型。在某些示例中，候选变量“ K”的数量可能比研究单元的数量“ N”大得多。这个想法已应用于各种统计模型，例如回归，图形模型，生存分析和聚类分析。尽管它很受欢迎，但理论特性，尤其是频繁的融合特性，尚未得到充分确立。最近，研究人员成功研究了BVS的常规线性线性模型的频繁收敛性（一致性，收敛速率和预测性能）。在该项目中，考虑了一个全新的方向，用于研究具有统计力学的吉布斯后验的BV。与通常的贝叶斯先验相反，它是根据似然函数构建的，吉布斯后部是由实际兴趣的风险函数（例如分类错误）构建的，旨在最大程度地限制风险功能而不对数据进行建模。这可以改善通常的贝叶斯方法的性能，因为通常的贝叶斯方法取决于可能被误指定的概率模型。研究人员研究了出于分类目的而构建的BVS的统计性能。提供条件，以便即使在具有高维度（KN）的情况下，BV也将达到良好的分类性能。 BVS具有多学科应用程序，包括各种数据挖掘实践，其中将从许多候选人中选择一些重要的变量，并用于预测和决策，例如模式识别，欺诈检测，国土安全性，面向客户的营销决策，机器学习，微阵列分析，微阵列分析和生物研究。这些应用程序典型地涉及许多候选变量（有时比样本量多得多）。通过选择一些重要的变量，BV可以对这些应用程序中的每个应用程序中的解释，预测和决策都非常有帮助，尽管潜在的维度可能很高。当前的项目将提供一个理论框架和条件，在某种意义上，在某种意义上，在某种意义上，具有Gibbs后部的BV将几乎是最佳的，因此为这项重要技术提供了理论上的理由。 BVS的扎实理论基础还将更好地解释从BVS获得的结果，并为从业者提供有关先前分布规范的有用信息。在许多情况下，如此良好的理论理解可能会导致经验绩效的改善。