Modified Cholesky Riemann Manifold Hamiltonian Monte Carlo: Exploiting Sparsity for Fast Sampling of High-dimensional Targets

Riemann manifold Hamiltonian Monte Carlo (RHMC) holds the potential for producing high-quality Markov chain Monte Carlo-output even for very challenging target distributions. To this end, a symmetric positive definite scaling matrix for RHMC, which derives, via a modified Cholesky factorization, from the potentially indefinite negative Hessian of the target log-density is proposed. The methodology is able to exploit sparsity, stemming from conditional independence modeling assumptions, of said Hessian and thus admit fast and highly automatic implementation of RHMC even for high-dimensional target distributions. Moreover, the methodology can exploit log-concave conditional target densities, often encountered in Bayesian hierarchical models, for faster sampling and more straight forward tuning. The proposed methodology is compared to Gibbs sampling and Euclidian metric Hamiltonian Monte Carlo on some challenging targets and illustrated by applying a state space model to real data.