Statistical Inference for Stochastic Processes, Analysis of MCMC Algorithms and Applications to Climate Science

这项建议由三个相辅相成的主题组成，特别侧重于对动力系统、偏微分方程式(通常代表物理现象)和扩散产生的数据进行统计推断。提案中的研究问题的动机是，在这些数据自然是高维的领域，确实需要新的统计推断。这一建议的一个突出特点是问题的跨学科性质，这需要综合应用数学、概率和统计等广泛领域的技术。主要内容包括1)高维马尔可夫链蒙特卡罗算法的稳定性，2)扩散和动力系统反问题的统计推断，以及3)在气候科学和温度预测中的应用。前两个主题旨在发展方法和提高对统计推断程序的理论理解，而第三个主题将直接落实从前两个主题中获得的见解，以回答气候科学中一些具体的、相关的和公开的问题。连接上述三个主题的研究主线是新的和高效的马尔可夫链蒙特卡罗技术的发展和理论分析。技术和计算能力的进步使得统计学中许多历史上难以解决的问题可以通过某些概率算法进行常规实现。尽管经过了二十年的密集研究，我们对这些复杂算法在高维空间的行为的理论理解仍然很原始。PI建议研究这些算法，并在高维上量化它们的行为，并将它们应用于解决气候科学中的具体问题。

This proposal consists of three complementary themes with a particular research focus on statistical inference for data arising from dynamical systems, partial differential equations (often representing physical phenomena) and diffusions. The research questions in the proposal are motivated by the genuine need for novel statistical inference in these areas where the data is naturally high dimensional. A highlight of this proposal is the interdisciplinary nature of the problems which requires the integration of techniques from a wide spectrum of fields in applied mathematics, probability and statistics. The main themes are 1) stability of Markov Chain Monte Carlo algorithms in high dimensions, 2) statistical inference for inverse problems from diffusions and dynamical systems, and 3) applications to climate science and temperature prediction. The first two themes aim at developing methods and improving the theoretical understanding of statistical inference procedures whilst the third will directly implement the insights gained from the first two to answer a few concrete relevant and open problems in climate science. The main thread connecting the above three themes of the research is the development and theoretical analysis of novel and efficient Markov Chain Monte Carlo techniques.Advances in technology and computing power have made many historically intractable problems in statistics amenable to routine implementation using certain probabilistic algorithms. Despite two decades of intense research, our theoretical understanding of the behavior of these complex algorithms in high dimensions is still primitive. The PI proposes to study these algorithms and quantify their behavior in high dimensions, and apply them to solve concrete problems in climate science.