A scale-free structure prior for Bayesian inference of Gaussian graphical models

The inference of gene association networks from gene expression profiles is an important approach to elucidate various cellular mechanisms. However, there exists a problematic issue that the number of samples is relatively small than that of genes. A promising approach to this problem will be to design regularization terms for characteristic network structures like sparsity and scale-freeness and optimize a scoring function including those regularization terms. The inference problem for gene association networks is often formulated as the problem of estimating the inverse covariance matrix of a Gaussian distribution from its samples. For this Bayesian inference problem, we propose a novel scale-free structure prior and devise a sampling method for optimizing a posterior probability including the prior. In a simulation study, scale-free graphs of 30 and 100 nodes are generated by the Barabási-Albert model, and the proposed method is shown to outperform another method which also use a scale-free regularization term. Our method is also applied to real gene expression profiles, and the resulting graph shows biologically meaningful features. Thus, we empirically conclude that our scale-free structure prior is effective in Bayesian inference of Gaussian graphical models.