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

所有富有成效的科学和统计分析都需要假设。一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施加的。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，无论是字面上的还是比喻上的。目前，统计模型在很大程度上依赖于过于精确的模型结构，这种结构建立在合理的科学知识和一些较难验证的假设的混合基础上。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，调查人员被迫编造既不能被信任也不能以有意义的方式宣布无效的高分辨率假设，这是当前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学家提供一个理论框架和实用方法来应对这一挑战，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关于不精确概率(IP)的大量文献，或者更广泛地说，概率和统计中的软方法(SMPS)。到目前为止，这两家公司都几乎没有受到统计界的关注，他们通常对任何不符合精确概率规则的事情都模棱两可。这个项目证明了精确概率和硬统计原理对研究IP和SMPS都有很大帮助，基本认识到一旦超出了精确概率，我们用来更新不精确模型的学习规则就必须成为隐含假设的工具，解释了IP和SMPS中出现的一些悖论和难题。随着对IP/SMPS能做什么和不能做什么的更清楚地理解，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。该项目的初始想法源于PI认识到，在处理低分辨率信息时，在缺失数据的文献中被广泛接受的Heitjan-Rubin数据粗化框架导致与Dempster-Shafer信任函数理论本质上相同的数学结构。因此，可以用普通概率来理解和研究信任函数。本研究旨在提供(1)信任函数的精确概率公式，它为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、收缩和确定丢失的现象；(3)探索高效的MCMC类算法的设计和实现，用于学习低分辨率推理的规则，并与MCMC并行用于贝叶斯推理。这项拟议研究的首要目标是增强科学家和统计学家的工具包，以进行更客观的推断和数据分析。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.