Compressible generalized hybrid Monte Carlo

Abstract

One of the most demanding calculations is to generate random samples from a specified probability distribution (usually with an unknown normalizing prefactor) in a high-dimensional configuration space. One often has to resort to using a Markov chain Monte Carlo method, which converges only in the limit to the prescribed distribution. Such methods typically inch through configuration space step by step, with acceptance of a step based on a Metropolis(-Hastings) criterion. An acceptance rate of 100% is possible in principle by embedding configuration space in a higher dimensional phase space and using ordinary differential equations. In practice, numerical integrators must be used, lowering the acceptance rate. This is the essence of hybrid Monte Carlo methods. Presented is a general framework for constructing such methods under relaxed conditions: the only geometric property needed is (weakened) reversibility; volume preservation is not needed. The possibilities are illustrated by deriving a couple of explicit hybrid Monte Carlo methods, one based on barrier-lowering variable-metric dynamics and another based on isokinetic dynamics.

Randomized Hamiltonian Monte Carlo

“…To correct the bias introduced by time discretization error, a Metropolis-Hastings accept-reject step is also added [27,15]. In order to keep the Metropolis-Hastings ratio simple, typically a volume-preserving and reversible method is used to numerically simulate Hamilton's equations in Step 2 [12]. The integrator of choice is the Verlet method, which is second-order accurate and, like Euler's rule, only requires one new evaluation of the gradient ∇Φ(q) per step.…”

Palindromic 3-stage splitting integrators, a roadmap

“…For integrators of the family (8), in general, ξ h = 1 + O(h 2 ) and θ h /h = 1 + O(h 2 ). The methods that satisfy (14) and have enhanced conservation of energy for quadratic problems (olive double-dot-dashed curve in Fig. 1) are precisely those with ξ h = 1 + O(h 4 ) (ellipses of low eccentricity).…”

Moment Conditions and Bayesian Non-Parametrics

“…Green ()) and compressible generalized hybrid Monte Carlo methods (in which the dynamics need not be volume preserving; see for instance Fang et al . ()). (In reversible jump MCMC sampling the chain is allowed to jump between models with a different number of parameters.…”