Advances and Applications in Bayesian Density Regression

在许多实证研究领域，回归模型经常被用来进行预测，以回答旨在推动研究和社会进步的问题。这种进步在很大程度上取决于准确的预测。然而，常用的回归模型可能会产生不准确的预测，因为它们做出了有问题的假设，而这些假设经常被数据违反。有问题的假设包括因变量与预测变量具有线性关系，因变量的方差不随预测变量而变化，以及因变量和随机预测变量的影响是正态分布的。本研究将探讨并发展一种贝叶斯非参数(BNP)回归模型，该模型允许因变量的整体分布(密度)随预测变量灵活且非线性地变化。BNP模型将由依赖于预测器的单峰分布的无限混合来定义，每个单峰分布由均匀分布的无限混合来建模。所有参数的先验分布将完成BNP模型的规范。BNP回归模型和该模型的多水平版本将被用来分析至少三个大数据集：(1)评估新的教师教育课程对学生基本技能的影响；(2)研究城市学校学生认为课文有意义的条件；(3)在元回归分析中研究反社会行为遗传性的预测因素。此外，将通过对各种数据生成条件下的许多模拟数据集的分析来评估BNP回归模型的性能。对于所有真实和模拟的数据集，预计BNP模型将显示出比其他常用回归模型更好的预测精度，并提供更多的科学见解。研究项目将通过开发和深入研究新的BNP回归模型来推动统计科学的发展。预计该模型的表现将优于目前用于预测的其他回归模型。通过对三个大型数据集的分析，这项研究将促进对重要社会问题的知识和科学理解，包括K-12数学、科学和扫盲教育的最佳实践，以及有情绪和行为障碍的青少年的治疗。最后，对于广大研究人员来说，该项目将为使用BNP回归模型进行数据分析提供用户友好的软件。这将有助于进一步促进对社会重要的研究问题的更准确的回答。

In many areas of empirical research, regression models often are used to make predictions in order to answer questions that aim to advance research and society. Such advancements hinge critically on accurate predictions. Commonly used regression models, however, can yield inaccurate predictions because they make questionable assumptions that often are violated by data. Questionable assumptions include that the dependent variable has linear relationships with the predictor variables, that the variance of the dependent variable does not change with the predictor variables, and that the dependent variable and random predictor effects are normally distributed. This research will investigate and develop a Bayesian nonparametric (BNP) regression model that allows the entire distribution (density) of the dependent variable to change flexibly and nonlinearly with the predictor variables. The BNP model will be defined by a predictor-dependent infinite mixture of unimodal distributions, with each unimodal distribution modeled by an infinite mixture of uniform distributions. A prior distribution on all parameters will complete the specification of the BNP model. The BNP regression model and a multi-level version of the model will be applied to analyze at least three large data sets: (1) to evaluate the effect of a new teacher education curriculum on the basic skills of its students; (2) to study the conditions under which urban school students find texts meaningful to read; and (3) to study the predictors of heritability of antisocial behavior in a meta-regression analysis. Furthermore, the performance of the BNP regression model will be evaluated through the analysis of many simulated data sets for a wide range of data-generating conditions. For all the real and simulated data sets, it is expected that the BNP model will show better predictive accuracy and will provide more scientific insights when compared with other regression models in common use.The research project will advance statistical science through the development and thorough investigation of a novel BNP regression model. This model is expected to outperform other regression models that currently are used for making predictions. Through the analysis of the three large data sets, the research will advance knowledge and scientific understanding of important societal issues, including best practices for K-12 math, science, and literacy education and the treatment of youth with emotional and behavioral disabilities. Finally, for a broad audience of researchers, this project will provide user-friendly software for performing data analysis with the BNP regression model. This will help further promote more accurate answers to research questions that are important to society.