Bayesian Inference for Relational Graph in a Causal Vector Autoregressive Time Series

We propose a method for simultaneously estimating a contemporaneous graph structure and autocorrelation structure for a causal high-dimensional vector autoregressive process (VAR). The graph is estimated by estimating the stationary precision matrix using a Bayesian framework. We introduce a novel parameterization that is convenient for jointly estimating the precision matrix and the autocovariance matrices. The methodology based on the new parameterization has several desirable properties. A key feature of the proposed methodology is that it maintains causality of the process in its estimates and also provides a fast feasible way for computing the reduced rank likelihood for a high-dimensional Gaussian VAR. We use sparse priors along with the likelihood under the new parameterization to obtain the posterior of the graphical parameters as well as that of the temporal parameters. An efficient Markov Chain Monte Carlo (MCMC) algorithm is developed for posterior computation. We also establish theoretical consistency properties for the high-dimensional posterior. The proposed methodology shows excellent performance in simulations and real data applications.