Bayesian Graphical Models under Positivity Constraints: A Scalable generalized likelihood Approach

2026-07-29

AI Summary

This work addresses the computational inefficiency in estimating precision matrices for high-dimensional fully positive Gaussian graphical models by proposing a generalized Bayesian framework based on the D-trace loss, which circumvents costly log-determinant computations. To induce sparsity, a spike-and-slab prior is incorporated. Three key techniques are introduced to enhance scalability and sampling efficiency in high dimensions: conditional independence–driven data augmentation, a fast matrix-normal sampler leveraging the Gram structure of the sample covariance, and an intertwining strategy that combines augmented and direct updates. Experimental results demonstrate that the proposed method achieves substantial computational acceleration on both synthetic and financial datasets while maintaining high estimation accuracy and superior graph structure recovery.

Abstract

We develop a computationally scalable Bayesian framework for precision matrix estimation in Gaussian graphical models under total positivity constraints. To overcome the high computational cost of the Gaussian likelihood, we adopt a generalized Bayesian approach based on the $D$-trace loss, which eliminates the log-determinant term and enables efficient optimization while allowing relaxation of positive definiteness during sampling. Sparsity is induced via spike-and-slab priors, and the resulting generalized posterior is shown to be proper under mild conditions. Our primary contribution is a suite of efficient posterior sampling algorithms tailored to high-dimensional settings. Starting from a component-wise Gibbs sampler, we introduce a novel data augmentation scheme that induces conditional independence among precision matrix entries, enabling joint updates. By exploiting the Gram structure of the sample covariance matrix, we further develop a fast matrix-normal sampler that significantly reduces per-iteration complexity in high-dimensional settings. An interweaving strategy combines augmented and direct updates to improve mixing without sacrificing scalability. Experiments on synthetic and financial data demonstrate substantial computational gains over existing methods, while maintaining competitive estimation accuracy and improved recovery of structured dependencies.

Problem

• Gaussian graphical models
• precision matrix estimation
• total positivity
• Bayesian inference
• high-dimensional statistics

Innovation

• generalized Bayesian inference
• D-trace loss
• spike-and-slab priors
• data augmentation
• matrix-normal sampler