Nonparametric Bayesian Modelling

非参数贝叶斯建模非参数贝叶斯模型的动机是希望更实际地建模数据。它们提供了一种逃避参数模型的限制的方法，同时，由于它们的贝叶斯性质，它们允许合并有关正在研究的过程的一些信息。这些哲学上的优势直接转化为模型的卓越性能，在涉及随机效应的情况下，这些模型的表现尤为突出。当这些模型与层次模型结合使用时，它们构成了一个强大的建模工具。目前，非参数贝叶斯建模的技术水平允许人们提出和拟合模型，并且在从一小部分候选模型中选择模型方面已经完成了有限的工作。在这种情况下，更大的数据分析领域几乎没有受到影响。这样做的原因是，当前的模型集不允许进行复杂的数据分析：提出模型，评估其拟合，开发模型的修改以提高其拟合，如果没有发现拟合的实质性改进，则减少模型的迭代过程。这项工作的主要重点是制定和实现复杂的非参数贝叶斯数据分析。为了实现上述目标，本研究计划确定了五个领域，其中非参数贝叶斯模型目前要么表现不佳，要么不能以令人满意的一般方式使用。这些领域是(i)协变量与响应之间关系的检验（ii）模型拟合性的评估（iii）相关实验信息的组合（iv）具有异常值的建模分布以及(v)小到中等样本量的工作。在提议的研究中，作者将开发对上述每个问题都表现出强大性能的模型。第一类模型，依赖非参数过程，超越了当前旨在提供单个随机分布描述的模型。它们提供了一种对表现出强局部依赖性和长期独立性的随机分布集合进行建模的方法。该模型的这一特性使其成为上述(i)至（iii）项的理想选择。第二类模型，污染模型，以一种清晰、易于解释的方式直接针对参数化模型的偏离。这种类型的模型对于描述包含异常值的分布是理想的，并且它与参数形式的接近性将产生几乎等同于参数模型的小样本性能，为问题（iv）和(v)提供了一种方法。此外，这两种类型的模型可以自由地组合在一起，从而产生一个单一的、一致的方法来解决所有五个问题。将研究模型的基本理论性质，设计拟合模型所需的计算策略，并将模型应用于各种设置。总体而言，研究的重点将放在发展良好的数据分析策略上，以便实现非参数贝叶斯建模的理论和实践优势。

Nonparametric Bayesian ModellingNonparametric Bayesian models are motivated by the desire to more realisticallymodel data. They provide a means of escaping the strictures of parametric models, while, with their Bayesian nature, they allow the incorporation of someinformation about the process under investigation. These philosophical advantages translate directly into superior performance for the models, whichhave had a particularly strong showing wherever random effects are involved.When these models are used in conjunction with the hierarchical model, they constitute a powerful modelling tool. The current state of the art in nonparametric Bayesian modelling allows one to propose and fit models, and a limited amount of work has been done on selectionof a model from some small set of candidate models. The greater field of dataanalysis in this context is almost untouched. The reason for this is that the current set of models do not allow for a sophisticated data analysis: the iterative process of proposing a model, assessing its fit, developing a modification of the model to improve its fit, and reducing the model if no substantial improvement in fit is found. The main focus of this work is to formulate and implement sophisticated nonparametric Bayesian data analysis. To accomplish the above goal, this research proposal identifies five areas where nonparametric Bayesian models currently either perform poorly or cannotbe used in a satisfactory, general fashion. These areas are (i) examinationof the relationship between covariates and response (ii) assessment of the fitof a model (iii) combination of information from related experiments (iv)modelling distributions with outliers and (v) working with small to moderatesample sizes. In the proposed research, the author will develop models that will show strongperformance for each of the above problems. The first class of models, dependent nonparametric processes, move beyond current models which are aimedat providing a description of a single random distribution. They provide a means of modelling a collection of random distributions which exhibit stronglocal dependence and long-range independence. This feature of the model makes it ideal for (i) - (iii) above. The second class of models, contaminated models, directly targets the departure from a parametric model in a clear, easily interpretable fashion. This type of model is ideal for describing distributions that contain outliers, and its close proximity to a parametric form will yield small sample performance that is nearly equivalent to the parametric model, providing an approach to problems (iv) and (v). Additionally,the two types of models may be freely combined, resulting in a single, coherent approach to all five problems. The basic theoretical properties of the modelswill be investigated, the computational strategies needed to fit the modelswill be devised, and the models will be applied in a variety of settings. Throughout, the emphasis of the research will be on the development of sounddata analytic strategies so that the theoretical and practical advantagesof nonparametric Bayesian modelling can be realized.