# Accelerating Bayesian Structure Learning in Sparse Gaussian Graphical Models

## TL;DR

This paper introduces a closed-form approximation for the ratio of G-Wishart normalizing constants, significantly accelerating Bayesian structure learning in sparse Gaussian graphical models while maintaining accuracy.

## Contribution

It presents a novel closed-form approximation for the G-Wishart ratio, enhancing the scalability and efficiency of Bayesian structure learning algorithms.

## Key findings

- The new method outperforms existing algorithms in computational speed.
- The approximation maintains high accuracy in structure learning.
- Implementation is available in the R package BDgraph.

## Abstract

Gaussian graphical models are relevant tools to learn conditional independence structure between variables. In this class of models, Bayesian structure learning is often done by search algorithms over the graph space. The conjugate prior for the precision matrix satisfying graphical constraints is the well-known G-Wishart. With this prior, the transition probabilities in the search algorithms necessitate evaluating the ratios of the prior normalizing constants of G-Wishart. In moderate to high-dimensions, this ratio is often approximated using sampling-based methods as computationally expensive updates in the search algorithm. Calculating this ratio so far has been a major computational bottleneck. We overcome this issue by representing a search algorithm in which the ratio of normalizing constant is carried out by an explicit closed-form approximation. Using this approximation within our search algorithm yields significant improvement in the scalability of structure learning without sacrificing structure learning accuracy. We study the conditions under which the approximation is valid. We also evaluate the efficacy of our method with simulation studies. We show that the new search algorithm with our approximation outperforms state-of-the-art methods in both computational efficiency and accuracy. The implementation of our work is available in the R package BDgraph.