Hybrid Monte Carlo on Hilbert spaces

Abstract

The Hybrid Monte Carlo (HMC) algorithm provides a framework for sampling from complex, high-dimensional target distributions. In contrast with standard Markov chain Monte Carlo (MCMC) algorithms, it generates nonlocal, nonsymmetric moves in the state space, alleviating random walk type behaviour for the simulated trajectories. However, similarly to algorithms based on random walk or Langevin proposals, the number of steps required to explore the target distribution typically grows with the dimension of the state space. We define a generalized HMC algorithm which overcomes this problem for target measures arising as finite-dimensional approximations of measures [pi] which have density with respect to a Gaussian measure on an infinite-dimensional Hilbert space. The key idea is to construct an MCMC method which is well defined on the Hilbert space itself. We successively address the following issues in the infinite-dimensional setting of a Hilbert space: (i) construction of a probability measure [Pi] in an enlarged phase space having the target [pi] as a marginal, together with a Hamiltonian flow that preserves [Pi]; (ii) development of a suitable geometric numerical integrator for the Hamiltonian flow; and (iii) derivation of an accept/reject rule to ensure preservation of [Pi] when using the above numerical integrator instead of the actual Hamiltonian flow. Experiments are reported that compare the new algorithm with standard HMC and with a version of the Langevin MCMC method defined on a Hilbert space.Hamiltonian dynamics Splitting technique Absolute continuity Hybrid Monte Carlo