2013/08/02 by Michela Ottobre, Ottobre, Michela, Natesh S. Pillai +6
Computer Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistical Methods and Inference #Theoretical and Computational Physics #math.PR
paper · pdf · doi:10.48550/arxiv.1308.0543
41 pages, 2 figures. This is the final version, with more comments and an extra appendix added
openalex publication_date 2013/08/02 · arxiv created 2014/04/03 · arxiv updated 2014/04/04 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We describe a new MCMC method optimized for the sampling of probability measures on Hilbert space which have a density with respect to a Gaussian; such measures arise in the Bayesian approach to inverse problems, and in conditioned diffusions. Our algorithm is based on two key design principles: (i) algorithms which are well-defined in infinite dimensions result in methods which do not suffer from the curse of dimensionality when they are applied to approximations of the infinite dimensional target measure on \bbRN; (ii) non-reversible algorithms can have better mixing properties compared to their reversible counterparts. The method we introduce is based on the hybrid Monte Carlo algorithm, tailored to incorporate these two design principles. The main result of this paper states that the new algorithm, appropriately rescaled, converges weakly to a second order Langevin diffusion on Hilbert space; as a consequence the algorithm explores the approximate target measures on \bbRN in a number of steps which is independent of N. We also present the underlying theory for the limiting non-reversible diffusion on Hilbert space, including characterization of the invariant measure, and we describe numerical simulations demonstrating that the proposed method has favourable mixing properties as an MCMC algorithm.