Abstract
Directional data emerges in a wide array of applications, ranging from atmospheric sciences to medical imaging. Modeling such data, however, poses unique challenges by virtue of their being constrained to non-Euclidean spaces like manifolds. Here, we present a unified Bayesian framework for inference on the Stiefel manifold using the Matrix Langevin distribution. Specifically, we propose a novel family of conjugate priors and establish a number of theoretical properties relevant to statistical inference. Conjugacy enables translation of these properties to their corresponding posteriors, which we exploit to develop the posterior inference scheme. For the implementation of the posterior computation, including the posterior sampling, we adopt a novel computational procedure for evaluating the hypergeometric function of matrix arguments that appears as normalization constants in the relevant densities.
Highlights
Analysis of directional data is a major area of investigation in statistics
There is a need for methodological development for a general sample space such as the Stiefel manifold (James, 1976; Chikuse, 2012) that goes beyond those techniques designed for simpler non-Euclidean spaces like the circle or the sphere
We explore several of its properties that are useful for subsequent theoretical development, and adopt an alternative parametrization of the matrix Langevin distribution so that the modified representation of the hypergeometric function can be used
Summary
Analysis of directional data is a major area of investigation in statistics. Directional data range from unit vectors in the simplest case to sets of ordered orthonormal frames in the general scenario. In early work, Mardia and Khatri (1977) and Jupp and Mardia (1980) investigated properties of the matrix Langevin distribution and developed inference procedures in the frequentist setup (Chikuse, 2012). We develop a comprehensive Bayesian framework for the matrix Langevin distribution, starting with the construction of a flexible class of conjugate priors, and proceeding all the way to the design of an practicable posterior computation. We propose two novel and reasonably large classes of conjugate priors, and based on theoretical properties of the matrix Langevin distribution and the hypergeometric function, we establish their propriety. We should note that a significant portion of the article is devoted to establishing a number of novel properties of the hypergeometric function of matrix arguments These properties play a key role in the rigorous development of the statistical procedures. We use matrix notation D in the place of d wherever needed, and vector d otherwise
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