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

贝叶斯建模和推理是常用的统计方法，用于分析来自许多科学领域的复杂高维数据，包括计算机视觉，生物学，生物统计学，生物信息学和医学。该研究项目涉及开发基于几何的，计算效率高且可扩展的工具，用于对具有揭示新见解的高潜力的数据集进行贝叶斯建模。一个例子是用于癌症中肿瘤异质性的统计分析的贝叶斯模型，其具有改进疾病表征和新治疗方法的可能性。这个项目的新奇和潜在的高影响力来自于在贝叶斯统计模型和推断的研究中称为微分几何的数学领域的实用性。虽然贝叶斯建模和推理领域在理论和计算方面都取得了很大进展，但很少有人注意研究这种模型的基本几何结构。在这个项目中，PI专注于为三个主要问题开发一个实用的，统一的黎曼几何框架：（1）贝叶斯灵敏度分析，（2）几何变分推理，（3）几何非参数先验构造;这些问题最终导致贝叶斯密度估计的第四个问题，其中前三个问题中描述的工具可以使用明显的优势。对于具有先验密度、抽样密度和后验密度的贝叶斯模型，通过平方根变换，将非参数Fisher-Rao度量下的概率密度非线性流形简化为欧氏度量下单位球面的正正交线，研究了模型的几何性质.因为球体的几何形状是众所周知的，所以用于分析的重要工具（例如，指数和反指数映射、并行传输、测地线）可以以封闭形式提供。因此，这个框架是通用的计算和适用于参数，半参数和非参数贝叶斯模型。更重要的是，它为定义密度之间的距离和开发几何校准措施提供了正式的数学背景。因此，该项目的两个主要贡献是（1）贝叶斯模型的基于度量的推理方法的发展，这可能允许对先验和后验信念进行更直观的解释，以及（2）通过对所有概率密度空间的内在分析，对后验推理的各个方面进行几何量化。

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.