Bayesian precision matrix estimation with applications to astrophysical data

Abstract

Estimation of a covariance matrix or its inverse, the precision matrix, is fundamental to multivariate analysis. Although the covariance matrix is more familiar to practitioners and more easily interpreted, it is often the precision matrix that is required. While many frequentist methods for precision matrix estimation have been proposed, and comprehensive reviews of recent developments are available, the same cannot be said of Bayesian methods for precision matrix estimation. This dissertation begins with a discussion of the progress in Bayesian precision matrix estimation that has led to interest in the connections between precision matrix estimation and univariate-response regression, followed by a review of mixturerepresentations of univariate densities that are important to the computational efficiency of the associated Markov chain Monte Carlo algorithms. The importance of Bayesian methods for precision matrix estimation to practitioners is then demonstrated in application to gravitational-wave interferometer data. Analysis of this data requires estimation of a precision matrix, which astrophysicists estimated by inverting an ill-conditioned estimate of the covariance matrix after thresholding the smallest eigenvalues. This approach is compared to direct estimation of the required precision matrix using Bayesian methods, which is demonstrated to have superior estimation performance. Finally, the regression framework for precision matrix estimation using the modified Cholesky decomposition is extended to the multivariate linear regression model. Using a novel regression prior that incorporates both sparse and dense settings and data augmentation, the proposed approach is demonstrated to be competitive with comparable methods and to offer distinct computational advantages compared to an alternative that also performs full posterior inference.