Gaussian Graphical Model Exploration and Selection in High Dimension Low Sample Size Setting

Gaussian graphical models (GGM) are often used to describe the conditional correlations between the components of a random vector. In this article, we compare two families of GGM inference methods: the nodewise approach and the penalised likelihood maximisation. We demonstrate on synthetic data that, when the sample size is small, the two methods produce graphs with either too few or too many edges when compared to the real one. As a result, we propose a composite procedure that explores a family of graphs with a nodewise numerical scheme and selects a candidate among them with an overall likelihood criterion. We demonstrate that, when the number of observations is small, this selection method yields graphs closer to the truth and corresponding to distributions with better KL divergence with regards to the real distribution than the other two. Finally, we show the interest of our algorithm on two concrete cases: first on brain imaging data, then on biological nephrology data. In both cases our results are more in line with current knowledge in each field.

Automated summary

In this paper, two families of Gaussian graphical model (GGM) inference methods, nodewise approach and penalised likelihood maximisation, are compared. The study demonstrates that when the sample size is small, both methods may result in graphs with either too few or too many edges compared to the real one. A composite procedure is proposed to address this issue, which explores a family of graphs with a nodewise numerical scheme and selects a candidate based on an overall likelihood criterion. This selection method yields graphs closer to the truth and corresponding to distributions with better KL divergence compared to the other two methods, particularly when the number of observations is small.