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Probabilistic Underpinning of Imprecise Probability and Statistical Learning with Low-Resolution Information

不精确概率的概率基础和低分辨率信息的统计学习

基本信息

批准号：
1812063
负责人：
Xiao-Li Meng
金额：
$ 19.99万
依托单位：
Harvard University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812063&HistoricalAwards=false
关键词：
Probabilistic Underpinning Imprecise Probability Statistical

项目摘要

All fruitful scientific and statistical analyses require assumptions. Some assumptions rightfully reflect past experience, present consensus, or future speculations. Others are imposed solely due to limitations of the investigation methods. Useful information in practice often comes in a vague, "low-resolution" form, like a blurred picture, both literally and figuratively. Currently, statistical models have largely relied on overly precise model structures, built upon a mix of sound scientific knowledge and some less verifiable assumptions. As models grow larger to accommodate the ever-growing volume and variety of data, statistical inference is faced with the pressing need to accurately and honestly express all types of low-resolution knowledge. Without adequate tools to deal with vague information, investigators are forced to concoct high-resolution assumptions that can neither be trusted nor invalidated in meaningful ways, the culprit in the ongoing crisis of irreplicable research. This project aims to provide scientists and statisticians both a theoretical framework and practical methods to tackle this challenge without having to abandon familiar probabilistic rules and tools, thereby strengthening the effort in reducing irreplicable scientific findings. The need to reduce unwanted assumptions in scientific and statistical studies has led to an extensive literature on imprecise probability (IP), or more broadly, soft methods in probability and statistics (SMPS). As of today, both have received little attention from the statistics community, which generally equivocates on anything that does not obey precise probabilistic rules. This project demonstrates that both the precise probability and hard statistical principles have much to offer for studying IP and SMPS, with the fundamental realization that once going beyond precise probabilities, the learning rules by which we update the imprecise model must become the vehicle for implicit assumptions, explaining some paradoxes and puzzles that arise in IP and SMPS. With a clearer understanding of what IP/SMPS can and cannot do, the proposed research contributes in theoretical and practical ways to ensure and enhance replicability of scientific studies that rely on probabilistic reasoning and statistical analysis.The initial idea of this project stemmed from the PI's realization that in handling low-resolution information, the well-accepted Heitjan-Rubin framework for data coarsening in the literature of missing data induces essentially the same mathematical structure as does the Dempster-Shafer theory of belief function. Consequently, belief function can be understood and studied using ordinary probability. The proposed research explores this link and extensions to its variations, and aims to provide (1) a precise probabilistic formulation of belief function, which offers both insights and questions for the Dempster-Shafer theory, especially Dempster's Rule of Combination; (2) a detailed comparison and contrast of three learning rules for updating and propagating low-resolution information, especially with respect to the phenomena of dilation, contraction, and sure loss; and (3) an exploration of the design and implementation of efficient, MCMC-type algorithms for learning rules of low-resolution inference, in parallel to MCMC for Bayesian inference. The overarching goal of the proposed research is to enhance the scientists' and statisticians' toolkit for conducting more objective inference and data analysis.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.

所有卓有成效的科学和统计分析都需要假设。有些假设正确地反映了过去的经验、现在的共识或未来的推测。另一些则完全是由于调查方法的限制而实施的。在实践中，有用的信息往往以模糊的、“低分辨率”的形式出现，就像一幅模糊的图片，无论是字面上还是比喻上。目前，统计模型在很大程度上依赖于过于精确的模型结构，建立在合理的科学知识和一些不太可验证的假设的基础上。随着模型变得越来越大，以适应不断增长的数据量和种类，统计推断面临着准确和诚实地表达所有类型的低分辨率知识的迫切需要。由于没有足够的工具来处理模糊的信息，研究人员被迫编造既不可信也不能以有意义的方式证明无效的高分辨率假设，这是持续的不可复制研究危机的罪魁祸首。该项目旨在为科学家和统计人员提供一个理论框架和实用方法，以应对这一挑战，而不必放弃熟悉的概率规则和工具，从而加强减少不可复制的科学发现的努力。在科学和统计研究中减少不必要的假设的需要导致了大量关于不精确概率（IP）的文献，或者更广泛地说，概率和统计（SMPS）中的软方法。到今天为止，这两个问题都没有得到统计界的关注，统计界通常对任何不遵守精确概率规则的事情都含糊其辞。该项目表明，精确概率和硬统计原理都可以为研究IP和SMPS提供很多东西，基本认识到一旦超越精确概率，我们更新不精确模型的学习规则必须成为隐含假设的工具，解释IP和SMPS中出现的一些悖论和难题。通过更清楚地了解IP/SMPS可以做什么和不可以做什么，拟议的研究在理论和实践方面做出了贡献，以确保和提高依赖于概率推理和统计分析的科学研究的可复制性。该项目的最初想法源于PI的认识，即在处理低分辨率信息时，在缺失数据的文献中，用于数据粗化的广为接受的Heitjan-Rubin框架本质上引入了与信任函数的Dempster-Shafer理论相同的数学结构。因此，信任函数可以用普通概率来理解和研究。本研究探讨了这一联系及其变化的扩展，旨在提供（1）一个精确的信念函数的概率公式，这为Dempster-Shafer理论，特别是Dempster的组合规则提供了见解和问题;（2）详细比较和对比了用于更新和传播低分辨率信息的三种学习规则，特别是关于膨胀现象，收缩，和肯定的损失;和（3）探索的设计和实施有效的，MCMC型算法学习规则的低分辨率推理，并行MCMC贝叶斯推理。该研究的总体目标是增强科学家和统计学家的工具包，以便进行更客观的推断和数据分析。该奖项反映了NSF的法定使命，并被认为值得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来支持。