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

贝叶斯推理是一种强大的统计推理方法，它允许统计学家和其他科学家将现有知识与新的数据样本结合起来，以更好地推断和决策。从后验分布中抽样的困难是在高维/大数据分析中广泛采用贝叶斯过程的最大障碍之一。需要一种快速准确的后验近似方法来辅助贝叶斯统计在高维问题中的实际实现。本研究项目将使用相关优化领域的思想来开发满足这些要求的贝叶斯后验逼近方法。该方法将在金融、市场科学、流行病学、生物学、医学科学等广泛领域得到应用。更具体地说，统计中需要在高维问题中使用后验近似方法：(a)产生更容易被马尔可夫链蒙特卡罗（MCMC）探索的近似，以及(b)从理论角度很好地理解。本项目将利用Moreau-Yosida近似以及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异同。本研究还将有助于马尔可夫链蒙特卡罗算法的理论分析，特别侧重于了解高维环境下MCMC算法的混合时间。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。在许多应用研究领域，包括生物医学研究、流行病学研究、市场营销科学和社会科学研究，变量选择在其中起着重要作用。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，并对相关科学问题获得新的见解。在教育方面，本研究的材料将成为受资助博士生博士论文的重要组成部分。该项目还将使PI能够使用相关的科学问题和数据集来丰富他的班级以及他的同事所教的其他班级的学生的学习经验。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高知名度会议上的介绍，广泛传播到科学界。

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.