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。由于1990年代S的进步，能够进行贝叶斯分析的计算成为可能。如果没有今天可用的强大计算工具，分层贝叶斯建模几乎不可能实现。术语分层建模通常指的是先验可以被分成更多的分层的想法。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。 贝叶斯方法的应用受到统计学、计算机科学和一般数据科学不同领域的应用的推动。机器学习理论，这使得广泛的 贝叶斯统计的应用，是数据科学中的一个重要领域。我提出的研究计划主要集中在贝叶斯非参数推理中的问题，这意味着对总体中的参数只假设了很小的限制。在贝叶斯统计中增加一些限制是必要的，尽管对总体做出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。我提议的研究还将研究某些限制对总体的影响，以发展做出推断的适当的贝叶斯非参数先验。 非参数贝叶斯方法的研究主要集中在类似于Dirichlet过程先验的对象的应用上，即非参数先验。本研究的目的是从理论和应用两个方面提高对该问题的认识。理论工作将发展使用非参数先验作为一种完全贝叶斯方法的方法，以解决诸如估计和检验统计假设等问题。这项研究还将考虑使用这些非参数先验的大样本行为。随着观察次数的增加，大样本调查会得出统计结果。 当变量可能剧烈波动时，本研究还将考虑大样本理论来估计时间序列中的参数。这些变量在金融、保险和环境研究的应用中经常遇到，作为表现出极端行为的扰动的模型。我对表现为随机行走的时间序列特别感兴趣，在这种序列中，每一步的大小都显示出极端行为。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，如估计未知人群的支持率。 此外，查明理论技术与实践之间的差距将推动统计研究的新领域，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，并找到方法来向实践者展示如何在现实生活中应用技术。

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.