New algorithms for consistent model selection beyond linear models

统计模型构建是科学发现的重要组成部分。大数据时代，高维数据频繁出现。 近年来，在线性模型、广义线性模型和审查数据模型的框架中存在高维特征的模型选择一直是一个非常活跃的研究领域。 该 PI 的目标是在贝叶斯计算框架内开发用于模型选择的新算法，该算法可针对高维问题进行扩展。 PI 通过与大气科学、遗传学和运动机能学领域的科学家合作来推动拟议的研究，旨在开发广泛适用于统计建模和数据分析的方法。 最近的大部分工作都集中在通过惩罚或正则化来缩小规模。 贝叶斯计算方法在广义上解释时，在统计中发挥着重要作用，包括模型选择和估计，但在高维统计中面临着理论复杂性和计算可扩展性方面的重要障碍。 该 PI 旨在开发一个理论框架，从频率论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。所提出的工作的一个重要部分是开发一种改进的吉布斯采样器来选择稀疏模型，在存在高维变量的情况下，该采样器比标准 MCMC 算法更具可扩展性。贝叶斯方法在非凸目标函数问题中特别有用，其中贝叶斯计算方法在性能上比直接优化更稳健。该项目中考虑的此类问题的主要应用是审查数据的分位数回归。除了模型选择之外，PI 还提出了一种用于截尾分位数回归的新估计方法，该方法有望提高计算和统计效率。同样重要的是，新方法可以轻松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以及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研讨会以及在统计科学领域广泛阅读的期刊上发表来适当传播。

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.