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Dimension-independent likelihood-informed MCMC

Author(s)
Law, Kody J.H.; Cui, Tiangang; Marzouk, Youssef M
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Abstract
Many Bayesian inference problems require exploring the posterior distribution of high-dimensional parameters that represent the discretization of an underlying function. This work introduces a family of Markov chain Monte Carlo (MCMC) samplers that can adapt to the particular structure of a posterior distribution over functions. Two distinct lines of research intersect in the methods developed here. First, we introduce a general class of operator-weighted proposal distributions that are well defined on function space, such that the performance of the resulting MCMC samplers is independent of the discretization of the function. Second, by exploiting local Hessian information and any associated low-dimensional structure in the change from prior to posterior distributions, we develop an inhomogeneous discretization scheme for the Langevin stochastic differential equation that yields operator-weighted proposals adapted to the non-Gaussian structure of the posterior. The resulting dimension-independent and likelihood-informed (DILI) MCMC samplers may be useful for a large class of high-dimensional problems where the target probability measure has a density with respect to a Gaussian reference measure. Two nonlinear inverse problems are used to demonstrate the efficiency of these DILI samplers: an elliptic PDE coefficient inverse problem and path reconstruction in a conditioned diffusion. Keywords: Markov chain Monte Carlo; Likelihood-informed subspace; Infinite-dimensional inverse problems; Langevin SDE; Conditioned diffusion
Date issued
2015-10
URI
http://hdl.handle.net/1721.1/114545
Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
Journal
Journal of Computational Physics
Publisher
Elsevier BV
Citation
Cui, Tiangang et al. “Dimension-Independent Likelihood-Informed MCMC.” Journal of Computational Physics 304 (January 2016): 109–137 © 2015 Elsevier Inc
Version: Author's final manuscript
ISSN
0021-9991
1090-2716

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