Optimal Bayesian Inference Under Shape Restrictions

在统计建模的许多情况下，建模中使用的函数的形状起着关键作用。突出的例子是气候变化下北极冰盖融化和海平面上升的趋势日益加剧。许多反问题，如反卷积或截尾估计，也会导致有关函数的形状限制。在估计这些量并量化其推论中的不确定性时，应考虑到这种形状限制。对增加趋势或类似形状限制的测试对于验证导致这种形状限制的理论也很重要。在这个项目中，结合先验信息和观测数据进行推断的贝叶斯方法将在形状受限模型的背景下开发。研究成果将应用于人们感兴趣的各个领域。这项拟议的研究，除了为回答相关的数学问题开发新的思想、方法和计算技术外，还将对气候变化、肿瘤大小监测和审查数据等各种应用的决策产生重大影响。研究成果将通过arxiv预印本、期刊出版物、在会议和各种机构的讲座以及通过专题课程传播。该软件将通过CRAN和Pi的网站免费开发和分发。国际和平协会高度致力于博士生的建议和促进多样性，特别是来自女性和代表性不足的群体。26名博士生已经毕业，目前有4名博士生与他共事。PI的NSF补助金还支持他的博士生参加会议。国际和平协会还通过他获得的会议支助赠款促进妇女和少数群体的代表性。总共有21名女性研究人员和4名来自代表性不足的团体和许多年轻的美国参与者得到了支持。国际和平研究所将继续促进与这项提议相关的研究的多样性。研究生资助将用于形状限制推理研究和为得到的公式编写计算机代码。从最大似然法的角度对形状约束推理进行了较好的研究，但贝叶斯方法的研究较少。在贝叶斯方法中，定性形状限制形式的附加信息可以自然地混合在先验信息中。相关函数中的不确定性可以用贝叶斯可信区域来量化，这些区域相对容易从后验抽样中获得。了解这类集合的频率覆盖率是很重要的。在这个方案中，将采用一种新的计算优势的基于“投影后验”的贝叶斯方法，这也将更容易从理论上进行分析。对于单变量和多变量的形状限制，都将发展适合于形状限制推断的先验，例如从阶跃函数和B-样条级数获得的先验，并将研究投影后验。将建立局部和全球后部收缩速率。在单调性或其他形状约束下，将得到回归或密度函数在某一点的贝叶斯可信区间的渐近频率复盖率。将使用重新校准步骤来调整覆盖范围以满足目标值。将开发形状限制的渐近最优和计算上有利的贝叶斯检验。结果将推广到其他类型的单变量形状限制，如凸性或对数凹性，以及回归、密度估计和生存分析中的多变量单调性和凸性设置。开发的方法将应用于包括气候变化和医学数据在内的各种背景下。这项拟议的研究可能会为贝叶斯方法在形状限制推理中开辟一条全新的道路，并协调形状限制下的贝叶斯和频率不确定性量化，并可能成为未来几年进一步发展的种子。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。

In many contexts of statistical modeling, the shape of a function used in modeling plays a key role. Prominent examples are increasing trend of the Arctic ice sheet melting and the rising sea levels under climate change. Many inverse problems such as deconvolution, or estimation under censoring also lead to shape restrictions on the concerned functions. While estimating these quantities and quantifying the uncertainty in their inference, such shape restrictions should be taken into consideration. Testing for an increasing trend or a similar shape restriction is also important for validating a theory leading to such a shape restriction. In this project, Bayesian methods, which combine prior information and observed data to make an inference, will be developed in the context of shape-restricted models. The results will be applied in various fields of interest. The proposed research, apart from developing new ideas, methods and computational techniques for answering related mathematical questions, will provide a significant impact on making decisions in various application such as climate change, tumor size monitoring, and censored data. Research findings will be disseminated through arXiv preprints, journal publications, talks in conferences and various institutions, and through special topics courses. The software will be developed and distributed for free through CRAN and PI's website. The PI is highly committed to doctoral student advising and promoting diversity, especially from women and underrepresented groups. Twenty-six doctoral students already graduated and four are currently working with him. The PI's NSF grants also supported his doctoral students to travel to conferences. The PI also has the track record of promoting the representation of women and minorities through the conference support grants he obtained. In total 21 female researchers and 4 from under-represented groups and many young U.S. participants were supported. The PI will continue promoting diversity in research related to this proposal. The graduate student support will be used on shape-restricted inference research and on writing computer codes for the resulting formulae. Shape restricted inference has been studied well from the maximum likelihood perspective, but Bayesian methods have been less developed. In the Bayesian approach, additional information in the form of the qualitative shape restriction may be naturally blended in the prior. Uncertainty in the concerned functions can be quantified by Bayesian credible regions, which are relatively easy to obtain from posterior sampling. The frequentist coverage of such sets is important to know. In this proposal, a new computationally advantageous Bayesian approach based on a ``projection posterior'' will be adopted, which will also be easier to analyze theoretically. Suitable priors for shape restricted inference such as those obtained from step functions and B-splines series will be developed for both univariate and multivariate shape restrictions, and the projection posterior will be studied. Local and global posterior contraction rates will be established. Asymptotic frequentist coverage of Bayesian credible intervals for a regression or density function at a point under monotonicity or other shape constraints will be obtained. A recalibration step will be used to adjust the coverage to meet a targeted value. Asymptotically optimal and computationally advantageous Bayesian tests for shape restrictions will be developed. Results will be extended to other types of univariate shape restrictions like convexity or log-concavity and to multivariate monotonicity and convexity settings in regression, density estimation, and survival analysis. The methods developed will be applied in diverse contexts including climate change and medical data. The proposed research may open up a completely new path for the Bayesian approach in shape-restricted inference and reconcile Bayesian and frequentist uncertainty quantification under shape restriction and may serve as a seed for further development in the years to come.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.