Optimal Bayesian Inference Under Shape Restrictions

在统计建模的许多上下文中，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导致的北极冰盖融化趋势加剧和海平面上升。许多逆问题，如反褶积或滤波下的估计，也会导致有关函数的形状限制。在估计这些量和量化其推断中的不确定性时，应考虑到这些形状限制。对于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，贝叶斯方法，结合先验信息和观测数据进行推理，将在形状限制模型的背景下发展。研究结果将应用于各个感兴趣的领域。除了开发新的思想、方法和计算技术来回答相关的数学问题外，拟议的研究将对气候变化、肿瘤大小监测和审查数据等各种应用中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、会议和各种机构的演讲以及专题课程传播。该软件将通过CRAN和PI的网站免费开发和分发。PI高度致力于为博士生提供建议和促进多样性，特别是女性和代表性不足的群体。已经有26名博士生毕业，目前有4名正在跟随他工作。PI的NSF资助也支持他的博士生去参加会议。PI还通过他获得的会议支持补助金，在促进妇女和少数民族的代表性方面取得了良好的记录。总共有21名女性研究人员和4名来自代表性不足的群体以及许多年轻的美国参与者得到了支持。PI将继续促进与该提案有关的研究的多样性。研究生支持将用于形状限制推理研究和为所得公式编写计算机代码。从极大似然的角度对形状限制推理进行了很好的研究，但贝叶斯方法还不够成熟。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中。相关函数的不确定性可以通过贝叶斯可信区域来量化，而贝叶斯可信区域相对容易从后验抽样中获得。了解这些集合的频率覆盖是很重要的。在本提案中，将采用一种新的基于“投影后验”的计算优势贝叶斯方法，这也将更容易从理论上进行分析。对于单变量和多变量形状限制，将开发从阶跃函数和b样条序列中获得的适合形状限制推理的先验，并研究投影后验。将建立局部和全局后缩率。在单调性或其他形状约束下，得到回归函数或密度函数在一点处的贝叶斯可信区间的渐近频率覆盖。将使用重新校准步骤来调整覆盖率以满足目标值。渐近最优和计算优势贝叶斯检验形状限制将被开发。结果将扩展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。所开发的方法将应用于各种情况，包括气候变化和医疗数据。本研究为贝叶斯方法在形状限制推理中的应用开辟了一条全新的道路，调和了形状限制下贝叶斯和频率不确定性量化的关系，并为今后贝叶斯方法的进一步发展奠定了基础。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。

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.