Modified Cholesky Riemann Manifold Hamiltonian Monte Carlo: exploiting sparsity for fast sampling of high-dimensional targets

Riemann manifold Hamiltonian Monte Carlo (RMHMC) has the potential to produce high-quality Markov chain Monte Carlo output even for very challenging target distributions. To this end, a symmetric positive definite scaling matrix for RMHMC is proposed. The scaling matrix is obtained by applying a modified Cholesky factorization to the potentially indefinite negative Hessian of the target log-density. The methodology is able to exploit the sparsity of the Hessian, stemming from conditional independence modeling assumptions, and thus admit fast implementation of RMHMC 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 straightforward tuning. The proposed methodology is compared to alternatives for some challenging targets and is illustrated by applying a state-space model to real data.