Markov Chain Monte Carlo from Lagrangian Dynamics

Shiwei Lan, Vasileios Stathopoulos, Babak Shahbaba, Mark Girolami

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


Hamiltonian Monte Carlo (HMC) improves the computational efficiency of the Metropolis–Hastings algorithm by reducing its random walk behavior. Riemannian HMC (RHMC) further improves the performance of HMC by exploiting the geometric properties of the parameter space. However, the geometric integrator used for RHMC involves implicit equations that require fixed-point iterations. In some cases, the computational overhead for solving implicit equations undermines RHMC’s benefits. In an attempt to circumvent this problem, we propose an explicit integrator that replaces the momentum variable in RHMC by velocity. We show that the resulting transformation is equivalent to transforming Riemannian Hamiltonian dynamics to Lagrangian dynamics. Experimental results suggest that our method improves RHMC’s overall computational efficiency in the cases considered. All computer programs and datasets are available online ( to allow replication of the results reported in this article.

Original languageEnglish (US)
Pages (from-to)357-378
Number of pages22
JournalJournal of Computational and Graphical Statistics
Issue number2
StatePublished - Apr 3 2015
Externally publishedYes


  • Explicit integrator
  • Hamiltonian Monte Carlo
  • Riemannian manifold

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Discrete Mathematics and Combinatorics


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