  One of the fundamental tasks of science is to find explainable relationships
between observed phenomena. One approach to this task that has received
attention in recent years is based on probabilistic graphical modelling with
sparsity constraints on model structures. In this paper, we describe two new
approaches to Bayesian inference of sparse structures of Gaussian graphical
models (GGMs). One is based on a simple modification of the cutting-edge block
Gibbs sampler for sparse GGMs, which results in significant computational gains
in high dimensions. The other method is based on a specific construction of the
Hamiltonian Monte Carlo sampler, which results in further significant
improvements. We compare our fully Bayesian approaches with the popular
regularisation-based graphical LASSO, and demonstrate significant advantages of
the Bayesian treatment under the same computing costs. We apply the methods to
a broad range of simulated data sets, and a real-life financial data set.
