Unconstrained Bayesian Model Selection on Inverse Correlation Matrices With Application to Sparse Networks

Bayesian statistical inference for an inverse correlation matrix is challenging due to non-linear constraints placed on the matrix elements. The aim of this paper is to present a new parametrization for the inverse correlation matrix, in terms of the Cholesky decomposition, that is able to model these constraints explicitly. As a result, the associated computational schemes for inference based on Markov Chain Monte Carlo sampling are greatly simplified and expedited. The Cholesky decomposition is also utilized in the development of a class of hierarchical correlation selection priors that allow for varying levels of network sparsity. An explicit expression is obtained for the volume of the elicited priors. The Bayesian model selection methodology is developed using a Reversible Jump algorithm and is applied to a dataset consisting of gene expressions to infer network associations.