  Riemannian manifold Hamiltonian Monte Carlo (RMHMC) is a powerful method of
Bayesian inference that exploits underlying geometric information of the
posterior distribution in order to efficiently traverse the parameter space.
However, the form of the Hamiltonian necessitates complicated numerical
integrators, such as the generalized leapfrog method, that preserve the
detailed balance condition. The distinguishing feature of these numerical
integrators is that they involve solutions to implicitly defined equations.
Lagrangian Monte Carlo (LMC) proposes to eliminate the fixed point iterations
by transitioning from the Hamiltonian formalism to Lagrangian dynamics, wherein
a fully explicit integrator is available. This work makes several contributions
regarding the numerical integrator used in LMC. First, it has been claimed in
the literature that the integrator is only first-order accurate for the
Lagrangian equations of motion; to the contrary, we show that the LMC
integrator enjoys second order accuracy. Second, the current conception of LMC
requires four determinant computations in every step in order to maintain
detailed balance; we propose a simple modification to the integration procedure
in LMC in order to reduce the number of determinant computations from four to
two while still retaining a fully explicit numerical integration scheme. Third,
we demonstrate that the LMC integrator enjoys a certain robustness to human
error that is not shared with the generalized leapfrog integrator, which can
invalidate detailed balance in the latter case. We discuss these contributions
within the context of several benchmark Bayesian inference tasks.
