Theory and Methods of Statistical Inference

现代技术简化了大型复杂数据的收集，这些数据被用来回答许多科学和工程领域的重要研究问题。  统计模型和方法是这项研究的重要组成部分，理解这些方法需要统计建模和推理理论的进展。拟议的研究计划旨在加深我们对统计领域知识基础的理解，并为开发新的分析方法提供框架。统计理论研究寻找各种科学问题背后的共性。统计科学的理论和应用之间的反馈循环是该学科最有趣和最重要的方面之一。 将特别强调开发基于似然函数的推理方法，因为它在贝叶斯推理和频率推理中都起着核心作用。  关于这两种推理模式在科学进步中的作用仍然存在争论。 《纽约时报》（2014 年 9 月 29 日）对这场辩论做了非常通俗易懂的概述。仔细研究统计推断的基本原理有助于为这场辩论提供信息。  我的研究计划还强调使用渐近展开来研究推理方法的数学特性，这是一种研究方法如何依赖于所分析数据集大小的技术。  对于无限量的数据，贝叶斯方法和频率论方法是一致的，但事实证明，它们在有限样本中的分歧可以借助渐近展开来查明。   在当前的技术领域，科学家和工程师可以获得的数据量几乎是无限的，但随着数据集规模的增加，用于帮助我们理解数据结构的数学模型的复杂性也随之增加。  这些模型用于总结问题的关键特征，对正在研究的科学假设进行推断，并对我们在类似情况下可能期望看到的情况进行预测。  当模型变得非常复杂，特别是涉及测量之间复杂的依赖性时，统计推断在计算和理论上都面临挑战。  在计算上，我们可能无法构建似然函数，并且在推论上，我们可能无法根据似然函数评估估计量的属性。因此，针对特定应用设计了许多似然函数的简化。拟议研究计划的一个主要重点是了解这些的理论特性，从而阐明计算需求如何与科学需求相互作用，以实现准确和有效的推理。

Modern technology has simplified the collection of large and complex sets of data, which are being used to answer important research questions in many fields of science and engineering.  Statistical models and methods are an essential part of this research, and understanding these methods requires progress on the theory of statistical modelling and inference. The proposed research program is intended to deepen our understanding of the intellectual foundations of the field of statistics and to provide a framework for developing new methods of analysis. Research in statistical theory looks for commonalities underlying a wide range of scientific problems. The feedback cycle between theory and applications of statistical science is one of the most interesting and important aspects of the subject. Particular emphasis will be placed on developing methods of inference based on the likelihood function, as it plays a central role in both Bayesian and frequentist inference.  There continues to be an ongoing debate about the role of these two modes of inference in scientific advances; a very accessible overview of the debate was featured in the New York Times (September 29, 2014). Careful study of the basic principles of statistical inference can help to inform this debate.  My research program also emphasizes the study of mathematical properties of inference methods using asymptotic expansions, a technique that studies how methods depend on the size of the data set being analysed.  With infinite amounts of data, Bayesian and frequentist methods agree, but it turns out that their disagreement in finite samples can be pinpointed with the help of asymptotic expansions.   In the current technological landscape, the amount of data available to scientists and engineers is nearly unlimited, but as the size of a set of data increases, so does the complexity of the mathematical models used to help us understand the structure in the data.  These models are used to summarize key features of a problem, to make inferences about scientific hypotheses under study, and to make predictions for what we might expect to see in similar circumstances.  When the models become very complex, and particularly involve complex dependencies among measurements, statistical inference faces challenges both computationally and theoretically.  Computationally, we may not be able to construct the likelihood function, and inferentially we may not be able to assess the properties of estimated quantities based on the likelihood function. As a result a number of simplifications of likelihood functions have been designed for particular applications. A major focus of the proposed research program is understanding the theoretical properties of these, thus illuminating how computational needs interact with scientific needs for accurate and efficient inference.