CAREER: Default Bayesian Methods for Nonparametric Problems

非参数问题的默认贝叶斯方法复杂数据的统计模型通常包含一个或多个无限维参数，如概率密度、回归函数或马尔可夫过程的转换密度。近十年来，创新蒙特卡罗方案的快速发展使得在这些复杂问题中计算贝叶斯过程成为可能。然而，由于高维性，很少有可能从现有信息中完全得出先验或主观的结论。我们需要的是为包含可用先验信息的无限维参数构建先验的一般策略，例如回归函数或密度函数的平滑性（可微性）或形状（单调性、凸性、单模性）。理想情况下，构建的先验应该在给定的问题中进行测试，以避免估计中可能出现的陷阱。一致性和收敛率等大样本特性是备受推崇的基准测试标准。在本研究中，研究者使用默认方法为选择问题构建先验分布，设计合适的后验计算算法，开发计算软件，调查结果过程的大样本行为，通过中等大样本的模拟研究支持理论和方法，并将新方法应用于几个有趣的数据集。该研究为贝叶斯方法学家提供了具有已知性能属性的先验目录，从而促进了贝叶斯方法在其他高维参数模型中的应用。数据的现代统计模型在各种各样的应用中，如数据挖掘、图像分析、生物识别、生物统计学、生物信息学、信号处理和金融，通常依赖于高维或无限维参数，如生存分布、概率密度、回归函数、马尔可夫链的转移密度等。成功分析这些数据提出了在有限参数模型分析中没有发现的挑战，并且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，研究者开发了理论、方法和计算算法，用于实现依赖于无限维参数的复杂统计模型的默认贝叶斯分析。这项研究通过高级课程的教学以及出版物和研讨会等通常的科学渠道进行传播。该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。对已知性能的有用先验进行了编目，并为各种领域的现成应用开发了用户友好的软件。因此，该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。

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.