Bayes and Empirical Bayes Model Selection

-----------------------------------------------------------------------提案编号：DMS 9803756 PI: Edward I. George Institution: University of Texas项目：Bayes和Empirical Bayes模型选择摘要：针对各种不同的环境，研究、发展和增强了用于模型选择的Bayes和Empirical Bayes方法。首先，针对典型线性模型设置，建立了最近发现的经验贝叶斯选择准则的理论频域风险性质。对于图形模型选择，频率主义者和贝叶斯方法之间的联系进行了研究，并用于激发新的经验贝叶斯方法。对于小波表示，开发了鲁棒的经验贝叶斯选择程序，以适应重尾噪声分布。对于贝叶斯CART建模，开发了新的结构化层次先验族。为了方便和增强模型搜索，开发了新的MCMC算法，该算法可以跨模型集而不是单个模型移动。这些算法使用新的转移核步骤，例如移植，可以快速遍历在这种情况下出现的各种多模态模型后验分布。这些算法还结合了随机加热和冷却来进一步增加运动。本研究最终关注的是在大型多变量数据集中发现系统结构的统计方法的发展。由于收集和分析此类数据的技术的爆炸式增长，这个普遍问题非常重要。事实上，如此庞大的数据集自然出现在国家关注的联邦战略领域，包括电信、生物技术和气候学。在这种情况下，标准的统计方法是在一些预先指定的、大的、灵活的潜在模型类别中搜索“有前途的”统计模型。一旦发现，挑战在于评估模型并得出有意义的推论。当潜在模型的类别很大时，这些任务可能特别困难，因为变量的数量通常很大。这项工作的主要重点将是发展新的方法来应对这些挑战，并扩大统计方法的范围。

----------------------------------------------------------------------- Proposal Number: DMS 9803756 PI: Edward I. George Institution: University of Texas Project: Bayes and Empirical Bayes Model Selection Abstract: Bayes and empirical Bayes approaches for model selection are studied, developed and enhanced for a variety of different settings. First of all, theoretical frequentist risk properties of recently discovered empirical Bayes selection criteria are established for the canonical linear model setting. For graphical model selection, connections between frequentist and Bayesian methods are investigated and used to motivate new empirical Bayes methods. For wavelet representations, robust empirical Bayes selection procedures are developed which accommodate heavy tailed noise distributions. For Bayesian CART modeling, new families of structured hierarchical priors are developed. To facilitate and enhance model search, new MCMC algorithms are developed which move across sets of models rather than single models. These algorithms use new transition kernel steps, such as transplantation, which can rapidly traverse the kinds of multimodal model posterior distributions that arise in this context. These algorithms also incorporate stochastic heating and cooling to further increase movement. This research ultimately concerns the development of statistical methods for discovering systematic structure in large multi-variable data sets. This general problem is of substantial importance because of the explosive growth in the technology to collect and analyze such data. Indeed, such large data sets occur naturally in federal strategic areas of national concern including telecommunications, biotechnology, and climatology. A standard statistical approach in such settings is to search for "promising" statistical models within some prespecified, large, flexible class of potential models. Once found, the challenge is to estimate the mod el and draw meaningful inference. These tasks can be especially difficult when the class of potential models is huge, as is typically the case the number of variables is large. The main thrust of this work will be to develop new methods to confront these challenges and broaden the scope of the statistical approach.