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北极冰盖融化和海平面上升的趋势。许多反问题，如反卷积，或估计下的删失也导致有关的功能的形状限制。在估计这些量并量化其推断中的不确定性时，应考虑此类形状限制。测试增加的趋势或类似的形状限制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，结合联合收割机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。其结果将应用于各种感兴趣的领域。除了开发新的想法，方法和计算技术来回答相关的数学问题外，拟议的研究还将对气候变化，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。研究结果将通过arXiv预印本、期刊出版物、会议和各种机构的讲座以及专题课程传播。该软件将通过CRAN和PI的网站免费开发和分发。PI高度致力于博士生咨询和促进多样性，特别是来自妇女和代表性不足的群体。26名博士生已经毕业，4名目前正在与他一起工作。PI的NSF赠款还支持他的博士生前往会议。PI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。共有21名女性研究人员和4名来自代表性不足的群体和许多年轻的美国参与者获得了支持。PI将继续促进与本提案相关的研究的多样性。研究生的支持将用于形状限制推理研究和编写计算机代码的公式。 形状限制推理已经从最大似然的角度得到了很好的研究，但贝叶斯方法发展较少。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中。有关函数的不确定性可以用贝叶斯可信域来量化，这是相对容易获得的后验抽样。知道这些集合的频率论覆盖率是很重要的。在该提案中，将采用一种新的基于“投影后验”的计算上有利的贝叶斯方法，这也将更容易进行理论分析。合适的先验形状限制推理，如从步骤函数和B-样条系列将开发为单变量和多变量的形状限制，投影后验将进行研究。将建立局部和全局后验收缩率。在单调性或其他形状约束下，将获得回归或密度函数在一点处的贝叶斯可信区间的渐近频率论覆盖。 将使用重新校准步骤来调整覆盖范围，以满足目标值。渐近最佳和计算上有利的贝叶斯测试的形状限制将被开发。结果将扩展到其他类型的单变量形状限制，如凸性或对数，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不同的背景，包括气候变化和医疗数据。拟议的研究可能会开辟一个全新的路径贝叶斯方法在形状限制的推理和调和贝叶斯和频率论的不确定性量化下的形状限制，并可能作为一个种子，为进一步发展在未来几年。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.