  Posterior sampling is a task of central importance in Bayesian inference. For
many applications in Bayesian meta-analysis and Bayesian transfer learning, the
prior distribution is unknown and needs to be estimated from samples. In
practice, the prior distribution can be high-dimensional, adding to the
difficulty of efficient posterior inference. In this paper, we propose a novel
Markov chain Monte Carlo algorithm, which we term graph-enabled MCMC, for
posterior sampling with unknown and potentially high-dimensional prior
distributions. The algorithm is based on constructing a geometric graph from
prior samples and subsequently uses the graph structure to guide the transition
of the Markov chain. Through extensive theoretical and numerical studies, we
demonstrate that our graph-enabled MCMC algorithm provides reliable
approximation to the posterior distribution and is highly computationally
efficient.
