Skip to Main Content
 

Global Search Box

 
 
 
 

Files

File List


Thesis.pdf (8.37 MB)

ETD Abstract Container

Abstract Header

A Geometric Framework for Modeling and Inference using the Nonparametric Fisher–Rao metric

Saha, Abhijoy
http://orcid.org/0000-0002-3368-0554
http://rave.ohiolink.edu/etdc/view?acc_num=osu1562679374833421

Abstract Details

2019, Doctor of Philosophy, Ohio State University, Statistics.
With rapid increase in the quantity and complexity of available data, we have embraced the idea of using probability models for gaining deeper insights into data generating mechanisms. Probability density functions (PDFs) form the crux of such models and provide a rigorous framework for statistical inference. However, the representation space of PDFs is infinite-dimensional and non-linear, and relevant statistical tools need to be developed for use on this restricted function space of PDFs accordingly. This dissertation focuses on building a unified geometric framework for analyzing PDFs, and subsequently defining efficient computational tools for their statistical analysis. Specifically, we use a Riemannian geometric framework based on the nonparametric Fisher–Rao metric on the manifold of PDFs. Under the square-root density representation, the manifold can be identified with the positive orthant of the unit hypersphere, and the Fisher–Rao metric reduces to the standard L^2 metric. We consider three different theoretical and applied statistical problems, all of which utilize the fundamental idea of exploiting such a Riemannian structure of PDFs to perform valid statistical analysis in an efficient manner. First, we demonstrate the utility of geometry-based approaches in a class of PDF approximation methods. We propose a geometric framework for variational inference in Bayesian models, where we formulate the task of approximating the posterior density based on the α-divergence function, instead of the classical Kullback-Leibler divergence function. We also propose a novel gradient-based algorithm for the variational problem and examine its properties. Through multiple simulations and real-data applications, we demonstrate the utility of the proposed geometric framework and algorithm on several Bayesian models. Second, we consider the idea of assessing model performance under misspecifications, which is intricately linked to the geometry of the space of PDFs, and can be used to evaluate the impact of various model assumptions. We study sensitivity of nonparametric Bayesian models for density estimation, based on Dirichlet-type priors, to perturbations of either the precision parameter or the base probability measure in the prior. To quantify the different effects of the perturbations of the parameters and hyperparameters in these models on the posterior, we define three geometrically-motivated global sensitivity measures. We validate our approach using multiple simulation studies, and consider the problem of sensitivity analysis for Bayesian density estimation models in the context of three real datasets that have previously been studied. Third, we develop a statistical technique to quantify brain tumor heterogeneity via smoothed voxel-value PDFs. This novel representation allows us to capture detailed tumor characteristics that were undetectable by former approaches. We develop tools for comparing tumor profiles across patients, which can be used in conjunction with standard clustering approaches. We also use a Bayesian model-based approach to consider the notion of cluster enrichment. Our analyses of The Cancer Genome Atlas (TCGA)-based glioblastoma multiforme dataset reveal two clusters of patients with marked differences in tumor morphology, genomic characteristics, and prognostic clinical outcomes. Finally, we close with a brief summary of our contributions and a discussion of open problems, which we plan to explore in the future.
Sebastian Kurtek (Advisor)
Catherine Calder (Committee Member)
Steven MacEachern (Committee Member)
Jennifer Sinnott (Committee Member)
186 p.
Statistics
nonparametric Fisher-Rao metric; Riemannian manifold; geometry; statistics; variational Bayes; sensitivity analysis; tumor heterogeneity

Recommended Citations

Citations

  • Saha, A. (2019). A Geometric Framework for Modeling and Inference using the Nonparametric Fisher–Rao metric [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1562679374833421

    APA Style (7th edition)

  • Saha, Abhijoy. A Geometric Framework for Modeling and Inference using the Nonparametric Fisher–Rao metric. 2019. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1562679374833421.

    MLA Style (8th edition)

  • Saha, Abhijoy. "A Geometric Framework for Modeling and Inference using the Nonparametric Fisher–Rao metric." Doctoral dissertation, Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1562679374833421

    Chicago Manual of Style (17th edition)

osu1562679374833421
762
© 2019, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.