A Synthesis of Objective Bayes Factors for Model Selection and Hypothesis Testing

目的贝叶斯测试和模型选择近年来成为一个重要的研究课题，原因是贝叶斯原理被广泛接受，但在贝叶斯框架下缺乏实用的测试和模型选择工具。在这个项目中，考虑到目前为止的主要贡献，研究人员开发了“综合贝叶斯因子”，并开发了大量新的理论、方法论和计算机软件，以达到日常实践的水平。对于最常见的检验，如学生t检验、F检验、拟合度检验、同方差检验或变量间相关性检验，制定了具有良好先验知识的实用而具体的准则。没有简单的解决办法：为了满足客观性和大小样本一致性的要求，贝叶斯因子不能像g-先验那样以共轭先验为基础。开发这些测试的主要工具是本征先验理论，它挖掘出与几种客观贝叶斯方法相关的隐含先验，如本征贝叶斯因子、分数贝叶斯因子、EP-先验、基于测试统计的贝叶斯因子等。其指导原则是接受客观贝叶斯检验，只有当相关先验是合理的，程序是一致的和客观的。在这个项目中，发展并证明了客观贝叶斯学生t检验、F检验、同方差检验、拟合优度检验和小样本贝叶斯因子近似--一种改进的BIC。波多黎各大学(UPR)是美国主要的教育机构之一，由代表人数不足的群体组成。因此，这个项目将自然而然地将多样性纳入NSF计划。圣胡安的里奥皮德拉斯校区是普遍定期审议大学系统中最大和最古老的校区。数学系隶属于自然科学学院，还有生物系、化学系、计算机科学系、物理学系、环境研究系和热带生态系统研究所。众所周知，统计在学院需要加强，无论是在开设的课程还是研究成果方面。贝叶斯统计在世界范围内正经历着一次重要的发展。普遍定期审议中的一些研究人员，特别是环境科学、生物学、医学校园和Mayaguez校园工程学院的研究人员对此很感兴趣。该项目是该大学贝叶斯统计发展的一个焦点，这一发展将影响该大学以及美国地质调查局等一些联邦机构。该项目还加强了数学博士项目和计算机科学博士项目，并将为一名博士生提供支持，该博士生的论文将直接在这项资助下进行开发。

Objective Bayesian testing and model selection has become an important research subject recently, due to the widely accepted Bayesian principle and lack of practical tools for testing and model selection under the Bayesian framework. In this project the investigator develops 'synthesis Bayes Factor', taking into account the main contributions so far, and develops substantial new theory, methodology and computer software to reach the level of everyday practice. Practical and specific guidelines with sound priors are developed for the most common tests, like the Student-t test, F-Test, Goodness of Fit, test for homoscesdasticity or test for correlation among variables. There is no easy solution: To comply with desiderata of Objectivity and Large and Small sample consistency, the Bayes Factors can not be based on Conjugate Priors, like g-priors. The main tool to develop these tests is the Theory of Intrinsic Priors, which digs out the implicit priors associated with several objective Bayesian methods, like Intrinsic Bayes Factors, Fractional Bayes Factors, EP-Priors, Bayes Factors based on Tests Statistics, etc. The guiding principle is to accept an Objective Bayesian Test, only when the associated prior is sensible and the procedure is consistent and objective. In this project, Objective Bayesian Student-t tests, F-Tests, Homoscesdasticity tests, Goodness of Fit test and a small sample Bayes Factor approximation, an improved BIC, are developed and justified. The University of Puerto Rico (UPR) is one of the major educational institutions in the USA, that is comprised of underrepresented groups. Thus this project will naturally integrates diversity into the NSF programs. The Rio Piedras (RP) Campus in San Juan is the largest and oldest campus of the UPR University System. The Department of Mathematics is affiliated to the College of Natural Sciences, along with the Departments of Biology, Chemistry, Computer Science, Physics, Environmental Studies and the Institute of Tropical Ecosystems Studies. It is notorious that Statistics needs reinforcement in the College, both in terms of courses given and research output. Bayesian Statistics is undergoing an important development worldwide. It is seen with interest by several researchers in the UPR, particularly in Environmental Sciences, Biology, the Campus of Medical Sciences and the Engineering School at the Mayaguez Campus. This project serves as a focal point to the development of Bayesian Statistics in the University, and this development will influence the university as well as some Federal Agencies as USGS. This project also strengthens the PhD program in Mathematics and the PhD program in Computer Science, and will serve as support of one Ph.D student whose thesis would be directly developed under this grant.