Date of Submission

12-22-2020

Date of Award

12-22-2021

Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Mathematics

Department

Interdisciplinary Statistical Research Unit (ISRU-Kolkata)

Supervisor

Bhattacharya, Sourabh (ISRU-Kolkata; ISI)

Abstract (Summary of the Work)

Inverse problems, where in a broad sense the task is to learn from the noisy response about some unknown function, usually represented as the argument of some known functional form, has received wide attention in the general scientific disciplines. However, apart from the class of traditional inverse problems, there exists another class of inverse problems, which qualify as more authentic class of inverse problems, but unfortunately did not receive as much attention.In a nutshell, the other class of inverse problems can be described as the problem of predicting the covariates corresponding to given responses and the rest of the data. Since the model is built for the responses conditional on the covariates, the inverse nature of the prediction problem is evident. Our motivating example in this regard arises in palaeoclimate reconstruction, where the model is built for the multivariate species composition conditional on climate; however, it is of interest to predict past climate given the modern species and climate data and the fossil species data. In the Bayesian context, it is natural to consider a prior for covariate prediction.In this thesis, we bring to attention such a class of inverse problems, which we refer to as ‘inverse regression problems’ to distinguish them from the traditional inverse problems, which are typically special cases of the former, as we point out. Development of the Bayesian inverse regression setup is the goal of this thesis. We particularly focus on Bayesian model adequacy test and Bayesian model and variable selection in the inverse contexts, proposing new approaches and illuminating their asymptotic properties.Towards Bayesian model adequacy, we adopt and extend the inverse reference distribution approach of Bhattacharya (2013), proving the convergence properties. Along the way, out of necessity, we develop asymptotic theories for Bayesian covariate consistency and posterior convergence theories of unknown functions modeled by suitable stochastic processes embedded in normal, double-exponential, binary and Poisson distributions that include rates of convergence and misspecifications.In the realm of inverse model and variable selection, we first develop an asymptotic theory for Bayes factors in the general setup, and then introduce pseudo-Bayes factors for model selection, showing that the asymptotic properties of the two approaches are in agreement, while the latter is more useful from several theoretical and computational perspectives. Along with the inverse regression setup we also develop the forward regression context, where the aim is to predict new responses given known covariate values, and illustrate the suitability, differences and advantages of the approaches, with various theoretical examples and simulation experiments. We further propose and develop a novel Bayesian multiple testing procedure for model and variable selection in the inverse regression setup, also exploring its elegant asymptotic properties. Our simulation studies demonstrate that this approach outperforms Bayes and pseudo-Bayes factors with respect to inverse model and variable selection.As an interesting application encompassing most of our developments, we attempt to evaluate if the future world is likely to experience the terrifying global warming projection that has perturbed the scientists and policymakers the world over. Showing that the question falls within the purview of inverse regression problems, we propose a novel nonparametric model for climate dynamics based on Gaussian processes and exploit our inverse regression methodologies to conclude that there is no real threat to the world as far as global warming is concerned

Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843864

Control Number

ISILib-TH471

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

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