Mathematical Sciences: Polya Trees for Nonparametric Bayesian Analysis

Polya树是一类介于更具体的Dirichlet过程和更一般的无尾分布之间的概率模型。该研究项目将调查Polya树是否为一些Dirichlet过程不能令人满意的问题提供了可行的解决方案。希望Polya树的通用性可以用来创建更准确地反映先验和后验观点的模型，同时避免全类无尾分布的计算复杂性。建议的研究集中在四个方面：1。通过Polya树分量的联合分布对光滑性、单调性和单模性的建模意见，2。3.通过混合Polya树建模预测分布。通过探索围绕参数模型的非参数模型来评估参数模型的鲁棒性；评估非参数模型的一致性和4。推广狄利克雷过程，为随机分布中质点的大小提供更灵活的先验。为现实世界的现象建立数学模型是统计学中一个永恒的问题。为了有用，模型必须足够复杂，以准确地表示现象的基本特征，但也必须在计算上易于处理。拟议的研究项目调查了一类被称为“Polya树”的模型，它推广了众所周知的、易于处理的、但有些有限的一类被称为“Dirichlet过程”的模型。希望Polya树能够比Dirichlet过程更准确地表示现实世界的现象，同时保持易于处理。这项工作将部分通过理论计算，部分通过计算机模拟，部分通过将Polya树应用于实际数据的例子进行。

Polya trees are a class of probability models intermediate between the more specific Dirichlet processes and the more general tailfree distributions. The proposed research project will investigate whether Polya trees provide workable solutions in some problems where Dirichlet processes are unsatisfactory. The hope is that the generality of Polya trees can be used to create models that reflect prior and posterior opinions more accurately while avoiding the computational complexity of the full class of tailfree distributions. The proposed research concentrates in four areas: 1. Modelling opinions of smoothness, monotonicity and unimodality through the joint distributions of the components of the Polya tree, 2. Modelling predictive distributions through mixtures of Polya trees, 3. Assessing the robustness of parametric models by exploring nonparametric models concentrating around parametric ones; assessing the consistency of the nonparametric models and 4. Generalizing the Dirichlet process to yield more flexible priors for the sizes of the point masses in the random distribution. A constant problem in statistics is constructing mathematical models for real world phenomena. To be useful the models must be sufficiently sophisticated to represent essential features of the phenomena accurately but must also be computationally tractable. The proposed research project investigates a class of models called "Polya trees" that generalize the well known, tractable, but somewhat limited class of models called "Dirichlet processes." The hope is that Polya trees can represent real world phenomena more accurately than Dirichlet processes while remaining tractable. The work will proceed partly through theoretical calculation, partly through computer simulation and partly through examples of Polya trees applied to real data.