Log-density gradient covariance and automatic metric tensors for Riemann manifold Monte Carlo methods

A metric tensor for Riemann manifold Monte Carlo particularly suited for non-linear Bayesian hierarchical models is proposed. The metric tensor is built from here proposed symmetric positive semidefinite log-density gradient covariance (LGC) matrices. The LGCs measure the joint information content and dependence structure of both a random variable and the parameters of said variable. The proposed methodology is highly automatic and allows for exploitation of any sparsity associated with the model in question. When implemented in conjunction with a Riemann manifold variant of the recently proposed numerical generalized randomized Hamiltonian Monte Carlo processes, the proposed methodology is highly competitive, in particular for the more challenging target distributions associated with Bayesian hierarchical models.