  A metric tensor for Riemann manifold Monte Carlo particularly suited for
non-linear Bayesian hierarchical models is proposed. The metric tensor is built
from symmetric positive semidefinite log-density gradient covariance (LGC)
matrices, which are also proposed and further explored here. The LGCs
generalize the Fisher information matrix by measuring the joint information
content and dependence structure of both a random variable and the parameters
of said variable. Consequently, positive definite Fisher/LGC-based metric
tensors may be constructed not only from the observation likelihoods as is
current practice, but also from arbitrarily complicated non-linear prior/latent
variable structures, provided the LGC may be derived for each conditional
distribution used to construct said structures. 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.
