Improving the efficiency of Markov Chain Monte Carlo algorithms is an active area of re-
search in statistics. I will start this talk by providing a brief overview of Hamiltonian Monte
Carlo (HMC), which improves the computational eciency of the Metropolis algorithm by
reducing its random walk behavior. This of course requires numerical simulation of Hamilto-
nian dynamics and costly evaluation of the gradient of the log density function. Next, I will
present our recent work on improving HMC by ``splitting" the Hamiltonian in a way that
allows much of the movement around the parameter space to be done at low computational
cost. I will then discuss Riemannian Manifold HMC (RMHMC), which further improves
HMC's performance by exploiting the geometric properties of the parameter space. The ge-
ometric integrator used for RMHMC however involves implicit equations that require costly
numerical analysis (e.g., fixed-point iteration). I will finish my talk by presenting our recent
work on developing an explicit geometric integrator that replaces the momentum variable in
RMHMC by velocity.
