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A metric tensor for Riemann manifold Monte Carlo particularly suited fornon-linear Bayesian hierarchical models is proposed. The metric tensor is builtfrom symmetric positive semidefinite log-density gradient covariance (LGC)matrices, which are also proposed and further explored here. The LGCsgeneralize the Fisher information matrix by measuring the joint informationcontent and dependence structure of both a random variable and the parametersof said variable. Consequently, positive definite Fisher/LGC-based metrictensors may be constructed not only from the observation likelihoods as iscurrent practice, but also from arbitrarily complicated non-linear prior/latentvariable structures, provided the LGC may be derived for each conditionaldistribution used to construct said structures. The proposed methodology ishighly automatic and allows for exploitation of any sparsity associated withthe model in question. When implemented in conjunction with a Riemann manifoldvariant of the recently proposed numerical generalized randomized HamiltonianMonte Carlo processes, the proposed methodology is highly competitive, inparticular for the more challenging target distributions associated withBayesian hierarchical models.

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