CAREER: Default Bayesian Methods for Nonparametric Problems

职业：非参数问题的默认贝叶斯方法

• 批准号: 0349111
• 负责人: Subhashis Ghoshal
• 金额: $ 40万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0349111&HistoricalAwards=false
• 关键词: CAREER; Default; Bayesian; Methods; Nonparametric

DEFAULT BAYESIAN METHODS FOR NONPARAMETRIC PROBLEMSStatistical models for complex data often contain one or moreinfinite-dimensional parameters such as a probability density, aregression function, or the transition density of a Markovprocess. The rapid development of innovative Monte-Carlo schemesin the last decade makes it possible to compute Bayesprocedures in these complex problems. However, because of the highdimensionality, it is seldom possible to completely elicit a priorsubjectively from the available information. What is needed is ageneral strategy for constructing priors for infinite-dimensionalparameters that incorporates available prior information, such assmoothness (differentiability) or shape (monotonicity, convexity,unimodality) of a regression function or density function.Ideally, the constructed prior should be tested in the givenproblem to avoid possible pitfalls in estimation. Large-sampleproperties such as consistency and rate of convergence arewell-respected benchmark test criteria. In this research theinvestigator constructs prior distributions for selectproblems using a default approach, devises suitable algorithms forcomputation of the posterior, develops software for computation,investigates the large sample behavior of the resulting procedures,supports the theory and methods via simulation studies withmoderately large samples, and applies the new methods to severalinteresting data sets. The research provides Bayesianmethodologists with a catalog of priors with known performanceproperties, thereby facilitating the application of Bayes methodsin other models with high-dimensional parameters.Modern statistical models for data in a wide variety ofapplications, such as data mining, image analysis, biometrics,biostatistics, bioinformatics, signal processing, and finance,often depend on high- or infinite-dimensional parameters such as survival distributions, probability densities, regressionfunctions, transition densities of Markov chains, and so on. Successfulanalysis of such data presents challenges not found in theanalysis of finite-parameter models, and requires the developmentof new statistical theory, methods and software. A non-subjective Bayesian method retains the advantages of the Bayesian paradigm without requiring a subjective prior elicitation. In thisresearch the investigator develops the theory, methods, andcomputational algorithms for implementing default Bayesian analyses ofcomplex statistical models depending on infinite-dimensionalparameters. The research is disseminated through the teachingof advanced courses and via the usual scientific channels ofpublications and seminars. The research provides newdata-analytic tools for solving problems arising in diverse fields. Useful priors with known performance are cataloged and user friendly software is developed for ready applications to diverse fields. Thus the research has a major impact on the conduct of science in a number of highly-relevantapplication areas.

用于非参数问题的缺省贝叶斯方法复杂数据的统计模型通常包含一个或多个有限维参数，如概率密度、回归函数或马尔可夫过程的转移密度。在过去的十年中，创新的蒙特卡罗方法的迅速发展使得计算这些复杂问题的贝叶斯过程成为可能。然而，由于高维性，很少可能从可用信息中完全主观地得出先验。需要一种通用的策略来构造无限维参数的先验，它包含了可用的先验信息，如回归函数或密度函数的分类(可微性)或形状(单调性、凸性、单峰性)。理想情况下，应在给定问题中检验所构造的先验，以避免估计中可能出现的陷阱。大样本属性，如一致性和收敛速度，是备受推崇的基准测试标准。在这项研究中，研究人员使用默认方法为选定问题构造先验分布，设计合适的后验计算算法，开发计算软件，研究结果程序的大样本行为，通过中等大样本的模拟研究来支持理论和方法，并将新方法应用于几个有趣的数据集。现代统计模型在数据挖掘、图像分析、生物统计学、生物统计学、生物信息学、信号处理、金融等领域的广泛应用中，往往依赖于高维或无限维马尔可夫链的生存分布、概率密度、回归函数、转移密度等参数。对这些数据的成功分析提出了在有限参数模型分析中找不到的挑战，需要开发新的统计理论、方法和软件。非主观贝叶斯方法保留了贝叶斯范式的优点，而不需要主观的先验启发。在这项研究中，研究人员发展了实现依赖于无限维参数的复杂统计模型的缺省贝叶斯分析的理论、方法和计算算法。这项研究通过教授高级课程以及通过出版物和研讨会等通常的科学渠道进行传播。这项研究为解决不同领域出现的问题提供了新的数据分析工具。具有已知性能的有用的前科被编目，用户友好的软件被开发出来，用于各种领域的现成应用。因此，这项研究对许多高度相关的应用领域的科学行为产生了重大影响。