Theory of statistical inference

现代技术简化了大而复杂的数据集的收集，这些数据正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，理解这些方法需要在统计建模和推理理论方面取得进展。拟议的研究方案旨在加深我们对统计领域知识基础的了解，并为开发新的分析方法提供一个框架。统计理论研究寻找广泛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这门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，因为这在贝叶斯和频数推理方法中具有中心作用。关于在科学进步中使用这些不同的推理模式的争论仍在继续。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。这项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的数据集的大小的技术。对于无限数量的数据，贝叶斯方法和频率方法是一致的，但事实证明，在有限样本中，它们的不一致可以通过渐近展开来精确定位。 在目前的技术环境中，科学家和工程师可以获得的数据量几乎是无限的，但随着一组数据的大小增加，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，阐明正在研究的科学假设，并对我们在类似情况下可能会看到的情况做出预测。当模型变得非常复杂，特别是涉及到非常大量的参数时，相对于我们可以收集的观测数量，需要新的理论来提供统计总结、推断和预测的信息。这项研究计划的一个主要重点是为这些发展做出贡献。 非常大和/或复杂的数据通常被用来设计能够提供良好预测的算法；这是机器学习和人工智能的一些分支的许多工作的重点。关于数据收集和保护的统计最佳做法在评估这些方法在人口中广泛使用的可靠性方面非常有用。与推理相关的统计概念对于提高这些算法的可解释性也非常有用。这个研究项目将强调统计思维在从数据中学习的重要性。

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 this has a central role in Bayesian and frequentist approaches to inference. There continues to be an ongoing debate about the use of these different modes of inference in scientific advances. Careful study of the basic principles of statistical inference can help to inform this debate. This 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 shed light on 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 in particular involve very large numbers of parameters, relative to the number of observations we can collect, new theory is needed to inform for statistical summaries, inferences and predictions. A major focus of this research program is contributing to these developments. Data that is very large, and/or complex, is often used to design algorithms that give good predictions; this is the focus of much work in machine learning and some branches of artificial intelligence. Statistical best practices around the collection, and protection, of data can be very useful in assessing the reliability of these methods for widespread use in the population. Statistical concepts relevant to inference can also be very useful to advance the explainability of these algorithms. This research program will emphasize the importance of statistical thinking in learning from data.