Prior distributions for structured semi-orthogonal matrices

TL;DR

This paper develops a Bayesian framework for structured semi-orthogonal matrices, introducing priors that incorporate sparsity and smoothness, enabling efficient posterior inference and applications to biological and oceanographic data.

Contribution

It proposes a novel approach to prior construction for structured semi-orthogonal matrices, facilitating tractable Bayesian inference with specific priors for sparsity and smoothness.

Findings

Priors enable effective modeling of structured matrices in real data.

Parameter-expanded MCMC achieves tractable posterior inference.

Applications demonstrate practical utility in biological and oceanographic contexts.

Abstract

Statistical models for multivariate data often include a semi-orthogonal matrix parameter. In many applications, there is reason to expect that the semi-orthogonal matrix parameter satisfies a structural assumption such as sparsity or smoothness. From a Bayesian perspective, these structural assumptions should be incorporated into an analysis through the prior distribution. In this work, we introduce a general approach to constructing prior distributions for structured semi-orthogonal matrices that leads to tractable posterior inference via parameter-expanded Markov chain Monte Carlo. We draw on recent results from random matrix theory to establish a theoretical basis for the proposed approach. We then introduce specific prior distributions for incorporating sparsity or smoothness and illustrate their use through applications to biological and oceanographic data.