Scalable Multiple Network Inference with the Joint Graphical Horseshoe

TL;DR

This paper introduces a scalable ECM algorithm for the Bayesian graphical horseshoe estimator and a novel joint estimator for multiple networks, enhancing computational efficiency and leveraging shared structures in high-dimensional network inference.

Contribution

It develops a scalable ECM algorithm for the graphical horseshoe and proposes a joint estimator for multiple networks, improving efficiency and shared information utilization.

Findings

ECM algorithm is more scalable than Gibbs sampling.

Joint estimator outperforms existing methods in shared network structure detection.

Method maintains accuracy while handling high-dimensional data.

Abstract

Network models are useful tools for modelling complex associations. If a Gaussian graphical model is assumed, conditional independence is determined by the non-zero entries of the inverse covariance (precision) matrix of the data. The Bayesian graphical horseshoe estimator provides a robust and flexible framework for precision matrix inference, as it introduces local, edge-specific parameters which prevent over-shrinkage of non-zero off-diagonal elements. However, for many applications such as statistical omics, the current implementation based on Gibbs sampling becomes computationally inefficient or even unfeasible in high dimensions. Moreover, the graphical horseshoe has only been formulated for a single network, whereas interest has grown in the network analysis of multiple data sets that might share common structures. We propose (i) a scalable expectation conditional maximisation (ECM) algorithm for obtaining the posterior mode of the precision matrix in the graphical horseshoe, and (ii) a novel joint graphical horseshoe estimator, which borrows information across multiple related networks to improve estimation. We show, on both simulated and real omics data, that our single-network ECM approach is more scalable than the existing graphical horseshoe Gibbs implementation, while achieving the same level of accuracy. We also show that our joint-network proposal successfully leverages shared edge-specific information between networks while still retaining differences, outperforming state-of-the-art methods at any level of network similarity.