  Although the block Gibbs sampler for the Bayesian graphical LASSO proposed by
Wang (2012) has been widely applied and extended to various shrinkage priors in
recent years, it has a less noticeable but possibly severe disadvantage that
the positive definiteness of a precision matrix in the Gaussian graphical model
is not guaranteed in each cycle of the Gibbs sampler. Specifically, if the
dimension of the precision matrix exceeds the sample size, the positive
definiteness of the precision matrix will be barely satisfied and the Gibbs
sampler will almost surely fail. In this paper, we propose modifying the
original block Gibbs sampler so that the precision matrix never fails to be
positive definite by sampling it exactly from the domain of the positive
definiteness. As we have shown in the Monte Carlo experiments, this
modification not only stabilizes the sampling procedure but also significantly
improves the performance of the parameter estimation and graphical structure
learning. We also apply our proposed algorithm to a graphical model of the
monthly return data in which the number of stocks exceeds the sample period,
demonstrating its stability and scalability.
