Optimal Bayesian Inference Under Shape Restrictions

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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.