High-Dimensional Bayesian Computations: The Moreau-Yosida Posterior Approximation

高维贝叶斯计算：Moreau-Yosida 后验近似

• 批准号: 1854545
• 负责人: Yves Atchade
• 金额: $ 22.19万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1854545&HistoricalAwards=false
• 关键词: Dimensional; Bayesian; Computations; Moreau; Yosida

Bayesian inference is a powerful statistical inference method that allows statisticians and other scientists to combine existing knowledge with new data samples for better inferences and decisions. The difficulty of sampling from posterior distributions is one of the biggest impediments to a wider adoption of Bayesian procedures in high-dimensional/big data analysis. There is a need for fast and accurate posterior approximation methods to assist with the practical implementation of Bayesian statistics in high-dimensional problems. This research project will use ideas from the related field of optimization to develop a Bayesian posterior approximation method that satisfies these requirements. The methodology will find applications in a wide-range of areas such as finance, marketing science, epidemiology, biology, medical sciences, and others.More specifically, there is a need in statistics for posterior approximation methods in high-dimensional problems that: (a) produce approximations that are easier to explore by Markov Chain Monte Carlo (MCMC), and (b) are well-understood from a theoretical viewpoint. This project will use the Moreau-Yosida approximation and related tools from optimization and variational analysis to develop a Bayesian posterior approximation method that satisfies the above two conditions. The research from this project will help clarify similarities and differences between optimization and simulation problems. This research will also contributes to the theoretical analysis of Markov Chain Monte Carlo algorithms, with the special focus on understanding the mixing time of MCMC algorithms in high-dimensional settings. The project will also address open problems in high-dimensional Bayesian variable selection and will develop some novel modeling and computational solutions. There are many applied research areas, including biomedical research, epidemiology, marketing science, and social science research, where variable selection plays an important role. Hence, results from this research will allow researchers in those areas to better handle available data and gain new insights into relevant scientific questions. On the educational side, the material from this research will form a key component of the doctoral dissertation of the Ph.D. students supported by this grant. The project will also enable the PI to use the related scientific problems and datasets to enrich the learning experience of students in his classes and possibly other classes taught by his colleagues. Furthermore, novel methodologies from this research will be widely disseminated to the scientific community through presentation of academic seminars as well as presentations at high-visibility conferences in statistical computing.

贝叶斯推断是一种强大的统计推断方法，允许统计学家和其他科学家将现有知识与新的数据样本相结合，以获得更好的推断和决策。从后验分布中采样的困难是在高维/大数据分析中更广泛采用贝叶斯过程的最大障碍之一。有一个快速和准确的后验近似方法，以协助高维问题的贝叶斯统计的实际实施的需要。本研究计画将利用最佳化相关领域的想法，发展出符合这些要求的贝氏后验近似法。该方法将在广泛的领域，如金融，营销科学，流行病学，生物学，医学科学，和others.More具体地说，有一个需要在高维问题的后验近似方法的统计：（a）产生近似，更容易探索的马尔可夫链蒙特卡罗（MCMC），和（B）是很好地理解从理论的角度来看。本项目将使用Moreau-Yosida近似以及来自最优化和变分分析的相关工具，开发满足上述两个条件的贝叶斯后验近似方法。该项目的研究将有助于澄清优化和仿真问题之间的相似性和差异。这项研究也将有助于马尔可夫链蒙特卡罗算法的理论分析，特别是在高维环境中的MCMC算法的混合时间的理解。该项目还将解决高维贝叶斯变量选择中的开放问题，并将开发一些新的建模和计算解决方案。有许多应用研究领域，包括生物医学研究，流行病学，营销科学和社会科学研究，变量选择起着重要作用。因此，这项研究的结果将使这些领域的研究人员能够更好地处理现有数据，并获得对相关科学问题的新见解。在教育方面，这项研究的材料将成为博士论文的关键组成部分。获得该补助金支持的学生。该项目还将使PI能够使用相关的科学问题和数据集，以丰富学生在他的课堂上的学习经验，并可能由他的同事教授的其他课程。此外，这项研究的新方法将通过学术研讨会的介绍以及在统计计算方面的高知名度会议上的介绍，广泛传播给科学界。