Imprecise Probability and Valid Statistical Inference

该研究项目将推进基于不精确概率的统计、数据驱动的不确定性量化的基础。社会、行为和经济科学家所面临的问题的复杂性使得直接的理论研究几乎是不可能的，因此进展在很大程度上依赖于数据分析和统计推断。然而，最近科学界的重复危机造成了对统计的困惑和不信任。造成复制危机的统计因素之一是缺乏坚实的统计基础。频率论和贝叶斯论这两大主流思想流派非常不同，但都依赖于精确的概率。然而，使用精确概率来根据数据对未知进行推断，在威胁可复制性的特定意义上已被证明是无效的。统计推断从精确概率到不精确概率的转变将产生广泛的积极影响，并在统计学和社会、行为和经济科学以外的领域创造新的研究机会。该项目将使用在线研究人员。一个传播结果的平台。将创建公开可用的软件。该项目还将为研究生和早期职业研究人员提供宝贵的培训和经验。这个研究项目的一个高级目标是创建一个基于不精确概率的统计推断的单一理论。本研究项目将侧重于一个框架，该框架使用可证明有效的、数据依赖的、不精确的（或非加性的）概率来量化不确定性并得出关于未知的推论。研究者将证明这整个框架可以用最简单的不精确概率模型之一来表达，即可能性度量，这种简单性在很多方面对从业者都有好处。研究者将证明，粗略地说，每一个精确的或保守的频率论程序对应于一个有效的不精确概率，充分建立统计推断中不精确概率的基本性质。研究者还将开发新的和强大的基于不精确概率的方法来解决两个一般的和具有挑战性的统计问题：高维结构学习和没有统计模型的推理。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project will advance the foundations of statistical, data-driven uncertainty quantification based on imprecise probabilities. The complexity of the problems faced by social, behavioral, and economic scientists makes direct theoretical investigations virtually impossible, so progress relies heavily on data analysis and statistical inference. However, the recent replication crisis in science has created confusion about and distrust in statistics. Among the statistical factors contributing to the replication crisis is the lack of a solid foundation of statistics. The two dominant schools of thought, frequentist and Bayesian, are very different, but both rely on precise probabilities. The use of precise probabilities for drawing inference about unknowns based on data, however, has been shown to be invalid in a specific sense that threatens replicability. The shift from precise to imprecise probabilities for statistical inference will have broad positive impacts and create new research opportunities in fields beyond statistics and the social, behavioral, and economic sciences. The project will use the online Researchers.One platform for dissemination of results. Publicly available software will be created. The project also will provide valuable training and experience to graduate students and an early-career researcher.A high-level goal of this research project is to create a single theory of statistical inference based on imprecise probabilities. This research project will focus on a framework that uses provably valid, data-dependent, imprecise (or non-additive) probabilities to quantify uncertainty and draw inferences about unknowns. The investigator will demonstrate that this entire framework can be cast in terms of one of the simplest imprecise probability models, namely, possibility measures, and this simplicity benefits practitioners in various ways. The investigator will prove that, roughly, every exact or conservative frequentist procedure corresponds to a valid imprecise probability, fully establishing the fundamental nature of imprecise probabilities in statistical inference. The investigator also will develop new and powerful imprecise probability-based methods for two general and challenging statistical problems: structure learning in high dimensions and inference without a statistical model.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.