A Geometric Approach to Bayesian Modeling and Inference with the Nonparametric Fisher-Rao Metric

Ohio State University, The, Columbus OH

Investigators

Abstract

Bayesian modeling and inference are commonly used statistical approaches to the analyses of complex high-dimensional data from many scientific fields including computer vision, biology, biometrics, bioinformatics and medicine. This research project is concerned with developing geometry-based, computationally efficient and scalable tools for Bayesian modeling of such datasets that have high potential for revealing novel insights. An example is a Bayesian model for statistical analysis of tumor heterogeneity in cancer with the possibility for improved disease characterization and new treatment approaches. The novelty and potential for high impact of this project come from the utility of an area of mathematics called differential geometry in the study of Bayesian statistical models and inferences. While much progress has been made in the area of Bayesian modeling and inference both in terms of theory and computation, little attention has been given to studying the underlying geometry of such models. In this project, the PIs focus on developing a practical, unified Riemannian-geometric framework for three main problems: (1) Bayesian sensitivity analysis, (2) geometric variational inference, and (3) geometric nonparametric prior construction; these problems culminate in a fourth one of Bayesian density estimation, wherein the tools described in the first three can be used with obvious advantages. For a Bayesian model with prior, sampling and posterior densities, the geometric properties of the model are investigated and exploited through a square-root transformation, under which the nonlinear manifold of probability densities endowed with the nonparametric Fisher-Rao metric simplifies to the positive orthant of the unit sphere endowed with the Euclidean metric. Because the geometry of the sphere is well-known, important tools for analysis (e.g., exponential and inverse exponential maps, parallel transport, geodesics) are available in closed-form. As a result, this framework is versatile computationally and applicable to parametric, semiparametric and nonparametric Bayesian models. More importantly, it provides a formal mathematical background for defining distances between densities and developing geometrically calibrated measures. Thus, the two main contributions of this project are the development of (1) metric-based inferential methods for Bayesian models that may permit a more intuitive explanation of prior and posterior beliefs, and (2) a geometric quantification of various aspects of posterior inference through intrinsic analysis on the space of all probability densities.