New algorithms for consistent model selection beyond linear models

统计模型的建立是科学发现的重要组成部分。大数据时代，高维数据频频涌现。在线性模型、广义线性模型和删失数据模型的框架下，存在高维特征的模型选择是近年来非常活跃的研究领域。PI的目标是在贝叶斯计算框架内开发新的模型选择算法，这些算法可扩展用于高维问题。PI通过与大气科学、遗传学和运动学科学家的合作来推动拟议的研究，旨在开发在统计建模和数据分析中广泛适用的方法。最近的许多工作都集中在通过惩罚或正规化来缩小。当广义解释时，贝叶斯计算方法在统计学中发挥着宝贵的作用，包括模型选择和估计，但在高维统计学中面临着重要的障碍，无论是在理论上的复杂性还是在计算的可扩展性方面。PI旨在开发一个理论框架，从频率学家的角度证明模型选择的一致性，这为为什么贝叶斯模型选择方法可以提供L0惩罚的渐近近似提供了有趣的见解。改进的Gibbs采样器用于稀疏模型的选择，在高维变量存在的情况下，比标准的MCMC算法具有更强的可扩展性。贝叶斯方法在具有非凸目标函数的问题中特别有用，在这些问题中，贝叶斯计算方法可以比直接优化在性能上更健壮。项目中考虑的此类问题的一个主要应用是删失数据的分位数回归。除模型选择外，PI还提出了一种新的截尾分位数回归估计方法，该方法在计算和统计上都是有效的。同样重要的是，新方法很容易适应其他估计方法难以处理的一般形式的审查。PI将通过与博士生合作并为本科生提供研究经验，继续将研究与教育结合起来。研究成果将通过会议和讲习班以及在广为阅读的统计科学期刊上发表而得到适当传播。

Statistical model building is an important part of scientific discovery. In the big data era, high dimensional data arise frequently. Model selection in the presence of high dimensional features in the framework of linear models, generalized linear models, and models with censored data has been a very active area of research in recent years. The PI aims to develop new algorithms for model selection, within a Bayesian computational framework, that are scalable for high dimensional problems. The PI motivates the proposed research through collaborations with scientists in atmospheric sciences, genetics, and kinesiology, and aims to develop methodologies that are broadly applicable in statistical modeling and data analysis. Much of the recent work has focused on shrinkage through penalization or regularization. Bayesian computational methods, when interpreted broadly, play a valuable role in statistics, including model selection and estimation, but face important hurdles in high dimensional statistics, both in theoretical intricacy and in computational scalability. The PI aims to develop a theoretical framework to demonstrate model selection consistency from the frequentist perspective, which offers interesting insights into why Bayesian model selection methods can provide an asymptotic approximation to the L0 penalty. An important part of the proposed work is the development of a modified Gibbs sampler in the selection of sparse models that is much more scalable than standard MCMC algorithms in the presence of high dimensional variables. The Bayesian methods are especially useful in problems with non-convex objective functions, where Bayesian computation methods can be more robust in performance than direct optimization. A primary application of such a problem considered in the project is quantile regression for censored data. In addition to model selection, the PI proposes a new estimation method for censored quantile regression that promises to be computationally and statistically efficient. Equally importantly, the new method adapts easily to general forms of censoring that other estimation methods have found difficult to handle. The PI will continue integrating research with education by working with PhD students and by providing research experiences for undergraduate students. The research output will be properly disseminated through conferences and workshops and through publication in widely read journals in statistical science.