Investigation of Bayes Procedures: Theory, Modeling, and Computation

贝叶斯分析是数据科学中广泛使用的一种估计、预测和解释技术。复杂大数据的新时代对贝叶斯统计提出了前所未有的挑战。这项研究项目从三个不同的角度解决了这些新的挑战。首先，研究人员将在贝叶斯统计框架下进行严格的数学分析，研究先验知识和科学结论之间的关系。其次，研究人员的目标是找到能够考虑现代大数据新特征的数据集建模的新方法。最后，研究人员打算通过发明既快速又理论上可靠的新算法来推动贝叶斯计算的极限。预计研究结果将对常规应用贝叶斯统计的领域产生积极影响，包括人口遗传学、天文学、计算机视觉、政治学、社会科学和动物学。贝叶斯分析是建模和计算的重要统计框架。然而，当遇到特定问题时，正确地应用贝叶斯程序并不是微不足道的。先验、似然和算法的选择都会影响从后验分布得出的最终结论。尽管贝叶斯分析在各个科学领域都得到了成功的应用，但如何进行贝叶斯推理仍然缺乏坚实的理论基础。这个项目的目标是开发一个关于最优贝叶斯推理的连贯理论。具体地说，研究人员将学习：1)贝叶斯理论：参数、非参数和高维模型中的最优后验收缩；2)贝叶斯建模：不带干扰参数的似然函数和贝叶斯边缘可交换网络分析；3)贝叶斯计算：变分推理的算法和统计性质；以及4)在单细胞RNA测序分析中的应用。

Bayesian analysis is a widely used technique in data science for estimation, prediction, and interpretation. The new era of complex and big data imposes unprecedented challenges to Bayesian statistics. This research project addresses these new challenges from three different perspectives. First, the investigator will study the relation between prior knowledge and scientific conclusion by conducting rigorous mathematical analysis in the framework of Bayesian statistics. Second, the investigator aims to find novel ways of modeling data sets that can take into account new features of modern big data. Finally, the investigator intends to push the boundary of Bayesian computation by inventing new algorithms that are both fast and theoretically sound. The results of the research are expected to have a positive impact in areas that apply Bayesian statistics on a routine basis, including population genetics, astronomy, computer vision, political science, social science, and animal science.Bayesian analysis is an important statistical framework for both modeling and computation. However, applying Bayesian procedures correctly when encountering a specific problem is non-trivial. The selections of prior, likelihood, and algorithm all influence the final conclusion drawn from a posterior distribution. Despite many successful applications of Bayesian analysis in various scientific areas, solid theoretical foundations on how to perform Bayesian inference are still lacking. The goal of this project is to develop a coherent theory on optimal Bayesian inference. Specifically, the investigator will study: 1) Bayesian theory: optimal posterior contraction in parametric, nonparametric and high-dimensional models; 2) Bayesian modeling: likelihood functions that are free of nuisance parameters and Bayesian edge-exchangeable network analysis; 3) Bayesian computation: algorithmic and statistical properties of variational inference; and 4) applications to single-cell RNA sequencing analysis.