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.