Nonparametric Bayesian Regression for Categorical Responses: Novel Methodology for Modeling, Inference and Applications

研究者开发了灵活的贝叶斯混合模型和相应的方法来推理一些具有有序分类响应的回归问题。更具体地说，开发了以下方法：二元响应的非参数混合回归；有序回归，包括多元有序响应和混合有序-连续响应；有序回归关系的动态建模以及生物测定剂量-反应研究的建模和风险评估。所有的方法都建立在非参数先验的基础上，包括一般的混合先验模型，以及由应用领域和模型的推理目标驱动的更结构化的形式。关键的研究活动包括灵活的回归建模，基于非参数先验的响应分布过程的协变量值或非参数混合模型的响应和协变量的联合分布。这类模型可以对回归关系和条件响应分布进行丰富的推理。因此，与标准参数模型相比，它们改善了模型拟合和预测性能，也相对于现有的贝叶斯半参数工作。对于所有正在开发的建模方法，研究者研究了相关的理论性质，模型规范，先验启发，马尔可夫链MonteCarlo后验仿真技术以及模型检查和比较。有序分类响应的回归问题————涉及在有序尺度上记录的响应变量和相关解释变量的数据————在生物医学、环境和社会科学的各个领域都至关重要。随着这些领域的研究人员收集越来越多的涉及有序响应的数据，特别是随着时间或时间和空间的变化，需要进行分析，以增强他们对潜在过程的理解。这激发了对能够容纳一般有序回归关系的足够丰富的统计模型的需求。这项研究的主要动机是扩大这些科学家可用的有序回归建模工具的目录，在扩展贝叶斯非参数领域的方法的过程中，贝叶斯统计的一个新兴领域。由于其通用性，在本研究项目下开发的统计方法在若干科学领域具有实质性应用的潜力。一个特别有前途的应用领域涉及自然选择形式估计的进化生物学问题，因为它将表型特征与有序适应度测量（如生存、成熟或繁殖成功）联系起来。在这种情况下，对适应度曲面的改进估计以及对其时间和/或空间演变的理解可以对所研究群体的有效决策产生影响。

The investigator develops flexible Bayesian mixture models and corresponding methods for inference for a number of regression problems with ordinal categorical responses. More specifically, methodology is developed for: nonparametric mixture regression for binary responses; ordinal regression, including multivariate ordinal responses and mixed ordinal-continuous responses; dynamic modeling for ordinal regression relationships; and modeling and risk assessment for bioassay dose-response studies with an ordinal classification. All the methods build from nonparametric priors, including both general mixture prior models as well as more structured forms as motivated by the application area and inferential objectives of the model. The key research activity involves flexible regression modeling based on either dependent nonparametric priors for the process of response distributions indexed by covariate values or nonparametric mixture models for the joint distribution of the response(s) and covariates. These classes of models enable rich inference for both the regression relationship and for the conditional response distribution. Hence, they improve model fit and predictive performance compared to standard parametric models, but also relative to existing Bayesian semiparametric work. For all the modeling approaches under development, the investigator studies relevant theoretical properties, model specification, prior elicitation, Markov chain MonteCarlo posterior simulation techniques, and model checking and comparison.Regression problems with ordinal categorical responses -- involving data on response variables recorded on an ordinal scale and on associated explanatory variables -- are of key importance in various fields of the biomedical, environmental and social sciences. As researchers from these fields collect more and more data involving ordinal responses, especially over time or time and space, the need for analyses that enhance theirunderstanding of underlying processes grows. This inspires the need for sufficiently rich statistical models that can accommodate general ordinal regression relationships. The primary motivation for this research is to expand the catalog of ordinal regression modeling tools available to such scientists, in the process expanding the methodology in the field of Bayesian nonparametrics, a burgeoning area of Bayesian statistics. Due to their generality, the statistical methods developed under this research project have the potential for substantive applications in several scientific fields. A particularly promising area of application involves evolutionary biology problems on estimation of the form of natural selection as it relates phenotypic traits to ordinal fitness measures, such as survival, maturity or reproductive success. For such settings, improved estimation of the fitness surface as well as understanding of its temporal and/or spatial evolution can have an impact on effective decision making for the population under study.