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

由于贝叶斯原理被广泛接受，而在贝叶斯框架下缺乏实用的测试和模型选择工具，客观贝叶斯测试和模型选择近年来成为一个重要的研究课题。在这个项目中，研究者开发了“综合贝叶斯因子”，考虑到目前为止的主要贡献，并开发了大量的新理论、方法和计算机软件，以达到日常实践的水平。针对最常见的检验，如Student-t检验、f检验、拟合优度检验、均方差检验或变量间相关性检验，制定了具有可靠先验的实用和具体指南。这并不是一个简单的解决方案：为了符合客观性和大小样本一致性的要求，贝叶斯因子不能基于共轭先验，如g先验。开发这些测试的主要工具是内在先验理论，它挖掘出与几种客观贝叶斯方法相关的隐含先验，如内在贝叶斯因子、分数贝叶斯因子、ep先验、基于测试统计的贝叶斯因子等。指导原则是接受客观贝叶斯检验，只有当相关的先验是合理的，过程是一致和客观的。在本项目中，开发并验证了客观贝叶斯学生t检验、f检验、均方差检验、拟合优度检验和小样本贝叶斯因子近似（改进的BIC）。波多黎各大学（UPR）是美国主要的教育机构之一，由代表性不足的群体组成。因此，该项目将自然地将多样性纳入NSF项目。圣胡安的里约热内卢Piedras （RP）校园是普遍定期审议大学系统中最大和最古老的校园。数学系隶属于自然科学学院，另外还有生物系、化学系、计算机科学系、物理系、环境研究系和热带生态系统研究所。众所周知，学院的统计学需要加强，无论是在课程设置方面还是在研究成果方面。贝叶斯统计在世界范围内正经历着重要的发展。普遍定期审议的几位研究人员，特别是环境科学、生物学、医学院和马亚圭斯校区工程学院的研究人员，感兴趣地看到了这一点。这个项目是贝叶斯统计在大学发展的焦点，这一发展将影响到大学以及一些联邦机构，如美国地质勘探局。该项目还将加强数学博士和计算机科学博士项目，并将作为一名博士生的支持，该博士生的论文将在该资助下直接发展。

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.