Bayesian Scalable Precision Factor Analysis for Gaussian Graphical Models

doi:10.1214/24-BA1461

Bayesian Scalable Precision Factor Analysis for Gaussian Graphical Models

2026-03-01

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Noirrit Kiran Chandra

DOI：10.1214/24-BA1461

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We propose a novel approach to estimating a multivariate Gaussian precision matrix that relies on decomposing them into a low-rank and a diagonal component. Such decompositions are very popular for modeling large covariance matrices as they admit a latent factor based representation that allows easy inference. The same is however not true for precision matrices due to the lack of computationally convenient representations which restricts inference to low-to-moderate dimensional problems. We address this remarkable gap in the literature by building on a latent variable representation for such decomposition for precision matrices. The construction leads to an efficient Gibbs sampler that scales very well to high-dimensional problems far beyond the limits of the current stateof-the-art. The ability to efficiently explore the full posterior space also allows easy assessment of model uncertainty. Exact zeros in the matrix encoding the underlying conditional independence graph are then determined via a novel posterior false discovery rate control procedure. A near minimax optimal posterior concentration rate for estimating precision matrices is attained by our method under mild regularity assumptions. We evaluate the method's empirical performance through synthetic experiments and illustrate its practical utility in datasets from two different application domains.

Keywords:

false discovery rate control

Gaussian graphical models

latent factor models

Markov chain Monte Carlo

posterior concentration

precision matrix estimation

scalable computation

Bayesian Scalable Precision Factor Analysis for Gaussian Graphical Models

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