Nonparametric Bayesian inference with single and multivariate random probability measures; heavy tailed time series.

我的研究主要集中在基于贝叶斯建模的问题解决上。贝叶斯分析的计算能力是在20世纪90年代取得的进步。如果没有当今强大的计算工具，分层贝叶斯模型的使用几乎是不可能实现的。术语分层建模通常指的是可以将先验分解为进一步的分层层的想法。计算机速度的进步导致了使用贝叶斯推理和频率推理的范式转变。******贝叶斯方法的应用是由统计学、计算机科学和一般数据科学的不同领域的应用驱动的。机器学习理论是数据科学的一个重要领域，它广泛应用了贝叶斯统计。我提出的研究计划主要集中在贝叶斯非参数推理的问题上，这意味着在总体中只假设关于参数的次要限制。在贝叶斯统计中，添加一些限制是必要的，尽管对人口的最小假设是选择非参数先验的一个影响因素。这些假设的一个例子是一般线性模型中创新的对称性。我提出的研究还将研究某些限制对人口的影响，以开发适当的贝叶斯非参数先验来进行推断。******非参数贝叶斯方法的工作集中在类似于狄利克雷过程先验的对象的应用，即非参数先验。本研究的目的是在理论和应用上推进知识的发展。理论工作将发展使用非参数先验作为完全贝叶斯方法的方法来解决诸如估计和检验统计假设等问题。本研究还将考虑使用这些非参数先验的大样本行为。大样本调查在观察数量增加时得出统计结果。******提出的研究也将考虑大样本理论在估计参数的时间序列中，当变量可能波动很大。这些变量在金融、保险和环境研究的应用中经常遇到，作为表现极端行为的扰动模型。我对表现得像随机漫步的时间序列特别感兴趣，其中每一步的大小都显示出极端的行为。在这些变量的大样本理论中使用的机制可以用于其他研究领域，例如估计未知种群的支持度。******此外，确定理论技术与实践之间的差距将推动统计研究的新领域，并对统计科学的进步具有重要意义。在这项研究中，将努力在这方面取得进展，并找到向从业者展示如何在现实生活中应用技术的方法。*****

My research concentrates largely on solving problems based on Bayesian modeling. The ability to perform the computations for Bayesian analysis was made possible by advances in the 1990's. Use of hierarchical Bayesian modeling had almost no chance to be carried out without the powerful computation tools available today. The term hierarchical modeling often refers to the idea that the prior can be split up into further hierarchy layers. Advancement in the speed of computers led to a paradigm shift in using both Bayesian and Frequentist inference. ******The applications of Bayesian methods are motivated by applications in different areas of Statistics, Computer Sciences, and in general Data Science. Machine learning theory, which makes extensive***use of Bayesian Statistics, is an important area in Data Science. My proposed research program focuses mainly on problems in Bayesian nonparametric inference, meaning that only minor restrictions are assumed about the parameters in a population. Adding few restrictions will be necessary in Bayesian Statistics, despite the fact that making minimal assumptions on population is an influential factor to choose nonparametric priors. An example of these assumptions is the symmetry of innovations in general linear models. My proposed research will also study the impact of certain restrictions on the population to develop proper Bayesian nonparametric priors in making inferences.******Work on nonparametric Bayesian methods concentrates on applications of objects similar to the Dirichlet process priors, namely nonparametric priors. The goal of this research is to advance the knowledge both in theory and application. The theoretical work will develop ways to use the nonparametric priors as a fully Bayesian approach to problems such as estimation and testing statistical hypothesis. This research will also consider the large sample behavior for the use of these nonparametric priors. The large sample investigation derives statistical results when the number of observations grow.******The proposed research will also consider the large sample theory in estimating parameters in time series when variables may fluctuate wildly. These variables are encountered frequently in applications in finance, insurance and environmental studies, as models for perturbations that exhibit extreme behavior. I am especially interested in time series that behave like a random walk, where the size of each step shows extreme behaviors. The machinery used in large sample theory of these variables can be used in other research areas such as estimating the support of an unknown population.******Furthermore, identifying the gaps between theoretical techniques and practice will drive new areas of statistical research and important in progress of Statistical sciences. In this research, efforts will be made to progress on this front and to find ways to show practitioners how to apply techniques in real life.*****