Conditional Path Sampling of SDEs and the Langevin MCMC Method

Stuart, Andrew M.; Voß, Jochen; Wilberg, Petter

doi:10.4310/cms.2004.v2.n4.a7

Cited by 73 publications

(69 citation statements)

References 10 publications

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“…The transition paths start at ∂A and terminate at ∂B, and hence they can be viewed as paths of a bridge process between s A and s B. In this perspective, our work is related to the conditional path sampling for SDEs studied in [SVW04,RVE05,HSVW05,HSV07]. In those works, stochastic partial differential equations were proposed to sample SDE paths with fixed end points.…”

Section: Introductionmentioning

confidence: 99%

Reactive trajectories and the transition path process

Nolen

2014

Probab. Theory Relat. Fields

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Abstract. We study the trajectories of a solution Xt to an Itô stochastic differential equation in R d , as the process passes between two disjoint open sets, A and B. These segments of the trajectory are called transition paths or reactive trajectories, and they are of interest in the study of chemical reactions and thermally activated processes. In that context, the sets A and B represent reactant and product states. Our main results describe the probability law of these transition paths in terms of a transition path process Yt, which is a strong solution to an auxiliary SDE having a singular drift term. We also show that statistics of the transition path process may be recovered by empirical sampling of the original process Xt. As an application of these ideas, we prove various representation formulas for statistics of the transition paths. We also identify the density and current of transition paths. Our results fit into the framework of the transition path theory by E and Vanden-Eijnden.

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Section: Introductionmentioning

confidence: 99%

Reactive trajectories and the transition path process

Nolen

2014

Probab. Theory Relat. Fields

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“…The simulation of diffusion bridges has received much attention over the past decade, see for instance the papers Elerian et al [11], Eraker [12], Roberts and Stramer [21], Durham and Gallant [10], Stuart et al [22], Beskos and Roberts [5], Beskos et al [3], Beskos et al [4], Fearnhead [13], Papaspiliopoulos and Roberts [20], Lin et al [17], Bladt and Sørensen [6], Bayer and Schoenmakers [2] to mention just a few. Many of these papers employ accept-reject-type methods.…”

Section: Diffusion Bridgesmentioning

confidence: 99%

Guided proposals for simulating multi-dimensional diffusion bridges

Schauer¹,

Meulen²,

Zanten³

2017

Bernoulli

131

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A Monte Carlo method for simulating a multi-dimensional diffusion process conditioned on hitting a fixed point at a fixed future time is developed. Proposals for such diffusion bridges are obtained by superimposing an additional guiding term to the drift of the process under consideration. The guiding term is derived via approximation of the target process by a simpler diffusion processes with known transition densities. Acceptance of a proposal can be determined by computing the likelihood ratio between the proposal and the target bridge, which is derived in closed form. We show under general conditions that the likelihood ratio is well defined and show that a class of proposals with guiding term obtained from linear approximations fall under these conditions.

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“…Other sampling algorithms incorporate information about the derivative of the logarithm of the target distribution to guide the Markov chain toward the target space where samples should be mostly concentrated. For instance, when the target density is differentiable, one can use Langevin-based algorithms where the mean of the Gaussian proposal density is replaced with one iteration of a preconditioned gradient descent algorithm as follows [20,22,[38][39][40][41]:…”

Section: Designing Efficient Proposals In Mh Algorithmsmentioning

confidence: 99%

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

Marnissi

Chouzenoux

Benazza-Benyahia

et al. 2018

Entropy

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Abstract:In this paper, we are interested in Bayesian inverse problems where either the data fidelity term or the prior distribution is Gaussian or driven from a hierarchical Gaussian model. Generally, Markov chain Monte Carlo (MCMC) algorithms allow us to generate sets of samples that are employed to infer some relevant parameters of the underlying distributions. However, when the parameter space is high-dimensional, the performance of stochastic sampling algorithms is very sensitive to existing dependencies between parameters. In particular, this problem arises when one aims to sample from a high-dimensional Gaussian distribution whose covariance matrix does not present a simple structure. Another challenge is the design of Metropolis-Hastings proposals that make use of information about the local geometry of the target density in order to speed up the convergence and improve mixing properties in the parameter space, while not being too computationally expensive. These two contexts are mainly related to the presence of two heterogeneous sources of dependencies stemming either from the prior or the likelihood in the sense that the related covariance matrices cannot be diagonalized in the same basis. In this work, we address these two issues. Our contribution consists of adding auxiliary variables to the model in order to dissociate the two sources of dependencies. In the new augmented space, only one source of correlation remains directly related to the target parameters, the other sources of correlations being captured by the auxiliary variables. Experiments are conducted on two practical image restoration problems-namely the recovery of multichannel blurred images embedded in Gaussian noise and the recovery of signal corrupted by a mixed Gaussian noise. Experimental results indicate that adding the proposed auxiliary variables makes the sampling problem simpler since the new conditional distribution no longer contains highly heterogeneous correlations. Thus, the computational cost of each iteration of the Gibbs sampler is significantly reduced while ensuring good mixing properties.

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Conditional Path Sampling of SDEs and the Langevin MCMC Method

Cited by 73 publications

References 10 publications

Reactive trajectories and the transition path process

Reactive trajectories and the transition path process

Guided proposals for simulating multi-dimensional diffusion bridges

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

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