Bayesian Mixture Models: Unified Theoretical Frameworks and MCMC Methods

不同种类的贝叶斯混合模型，如有限混合模型、有限和无限隐马尔可夫模型、狄利克雷过程模型和密度分析模型，以及它们的推理方法传统上是从孤立的角度发展起来的。很少有人试图从一个共同的立场来看待这些模型。对于在现实世界的研究中越来越多地遇到的大量数据集，这些模型的推理技术通常需要大量的计算负担，从而排除了完全贝叶斯解决方案。提出的研究将(i)通过制定混合模型的一般类别来实现统一，这些混合模型的特殊情况是许多常见的混合模型；（ii）从共同的角度探索广义混合模型的重要理论性质，如后验一致性，并发现渐近性，这些渐近性构成了广泛适用和具有成本效益的推理策略的基础。（iii）开发有效的马尔可夫链蒙特卡罗技术，用于将广义混合模型拟合到大型数据集上；（iv）开发用户友好的统计软件，实现这些方法，目标是将它们传播给研究人员；(v)应用所提出的方法来分析公开可用的、高通量的比较基因组杂交（CGH）关于各种癌症的数据。贝叶斯混合模型在统计应用中无处不在，因为它们能够通过相对简单的结构捕捉现实世界的复杂性。这些模型在计算机科学、流行病学、经济学、金融、林业、遗传学和市场营销等不同领域都有应用。这一快速发展领域的应用范围在过去十年中呈爆炸式增长。通过其理论和方法的组成部分，本研究将建立不同类别的混合模型共享的关键特征。它将使混合模型在应用程序中的使用成为可能，如果不是不可能的话，目前由于数据量太大而难以拟合模型。研究者将确保通过开放获取软件开发、在主要期刊上发表、将建议的方法应用于基于微阵列的癌症数据分析以及在统计和主题会议上展示结果等方式有效传播研究成果。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)Different kinds of Bayesian mixture models, such as finite mixtures, finite and infinite hidden Markov models, Dirichlet process models and analysis of densities models, and their methods of inference have traditionally been developed from isolated perspectives. There have been few attempts to view these models from a common standpoint. For the massive data sets increasingly encountered in real world studies, the inferential techniques for these models often necessitate heavy computational burdens that preclude fully Bayesian solutions. The proposed research will (i) achieve a unification by formulating general classes of mixture models whose special cases are many common mixture models, (ii) explore from a common standpoint important theoretical properties of generalized mixture models, such as posterior consistency, and discover asymptotics that form the basis of broadly applicable and cost-effective inferential strategies, (iii) develop efficient Markov chain Monte Carlo techniques for fitting generalized mixture models to large datasets, (iv) develop user-friendly statistical software that implement these methods with the goal of disseminating them to researchers, and (v) apply the proposed methods to analyze publicly available, high-throughput Comparative Genomic Hybridization (CGH) data on various kinds of cancer.Bayesian mixture models are ubiquitous in statistical applications because of their ability to capture real-world complexities through relatively simple constructions. These models have found application in such diverse areas as computer science, epidemiology, economics, finance, forestry, genetics, and marketing. The range of applications of this rapidly developing area has exploded in the last decade. Through its theoretical and methodological components, this research will establish key characteristics shared by disparate classes of mixture models. It will enable the utilization of mixture models in applications where it is currently difficult, if not impossible, to fit the models due to sheer volume of data. The investigator will ensure effective dissemination of the research through open-access software developments, publication in leading journals, application of the proposed methods to the analysis of microarray-based cancer data, and presentation of the results in statistical and subject-matter conferences.