Parallel Markov chain Monte Carlo Simulation by Pre-Fetching

Brockwell, Anthony

doi:10.1198/106186006x100579

Cited by 96 publications

(82 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is based on the observation that new (future) configurations in the Markov chain can be constructed on the fly and instantaneously, before the energy of the current configuration has been evaluated, as neither energy nor forces are required for the MC moves. 55 This approach requires speculation on the outcome of the acceptance check, or when used in a systematic fashion, builds a tree of configurations assuming both possible outcomes of this check. All the configurations present in the tree can be computed simultaneously, limited only by the available resources.…”

Section: Methodsmentioning

confidence: 99%

Dielectric Properties of Water Ice, the Ice Ih/XI Phase Transition, and an Assessment of Density Functional Theory

et al. 2014

View full text Add to dashboard Cite

The dielectric properties of the hydrogen disordered hexagonal phase (Ih) of water ice have been computed using density functional theory (DFT) based Monte Carlo simulations in the isobaric isothermal ensemble. Temperature dependent data yield a fit for the Curie-Weiss law of the system and hence a prediction of the temperature of the phase transition from the Ih phase to the hydrogen ordered ice XI phase. Direct simulations around the phase transition temperature confirm and refine the predicted phase transition temperatures and provide data for further properties, such as the linear thermal expansion coefficient. Results have been obtained with both hybrid and semilocal density functionals, which yields insight in the performance of the electronic structure method. In particular, the hybrid functional yields significantly more realistic dielectric constants than the semilocal variant, namely epsilon approximate to 116 as opposed to epsilon approximate to 151 at 273 K (epsilon(experiment) = 95). This can be attributed to the tendency of semilocal functionals to be biased to configurations with a large dipole moment, and their overestimation of the dipole moments of these configurations. This is also reflected in the estimates of the Ih/XI transition temperature, which is 70-80 and 90-100 K for the hybrid and semilocal functional respectively. DFT based sampling of the millions of configurations necessary for this work has been enabled by a Tree Monte Carlo algorithm, designed for massively parallel computers. with both hybrid and semi-local density functionals, which yields insight in the performance of the electronic structure method. In particular, the hybrid functional yields significantly more realistic dielectric constants than the semi-local variant, namely ε ≈ 116 as opposed to ε ≈ 151 at 273K (ε experiment = 95). This can be attributed to the tendency of semi-local functionals to be biased to configurations with a large dipole moment, and their overestimation of the dipole moments of these configurations. This is also reflected in the estimates of the Ih/XI transition temperature, which is 70-80K and 90-100K for the hybrid and semi-local functional respectively.DFT based sampling of the millions of configurations necessary for this work has been enabled by a Tree Monte Carlo algorithm, designed for massively parallel computers.

show abstract

Section: Methodsmentioning

confidence: 99%

Dielectric Properties of Water Ice, the Ice Ih/XI Phase Transition, and an Assessment of Density Functional Theory

et al. 2014

View full text Add to dashboard Cite

show abstract

“…Algorithms such as Multiple Try Metropolis (20), Ensemble MCMC (21), and "prefetching" approaches (22,23) all allow the computation of multiple proposals or future proposed paths in parallel, although only one proposal or path is subsequently accepted by the Markov chain, resulting in wasted computation of the remaining points. Another class of approaches involves incorporating rejected states into the Monte Carlo estimator with an appropriate weighting, such that the resulting estimate is still unbiased.…”

mentioning

confidence: 99%

A general construction for parallelizing Metropolis−Hastings algorithms

Calderhead

2014

Proc. Natl. Acad. Sci. U.S.A.

120

130

View full text Add to dashboard Cite

Markov chain Monte Carlo methods (MCMC) are essential tools for solving many modern-day statistical and computational problems; however, a major limitation is the inherently sequential nature of these algorithms. In this paper, we propose a natural generalization of the Metropolis−Hastings algorithm that allows for parallelizing a single chain using existing MCMC methods. We do so by proposing multiple points in parallel, then constructing and sampling from a finite-state Markov chain on the proposed points such that the overall procedure has the correct target density as its stationary distribution. Our approach is generally applicable and straightforward to implement. We demonstrate how this construction may be used to greatly increase the computational speed and statistical efficiency of a variety of existing MCMC methods, including Metropolis-Adjusted Langevin Algorithms and Adaptive MCMC. Furthermore, we show how it allows for a principled way of using every integration step within Hamiltonian Monte Carlo methods; our approach increases robustness to the choice of algorithmic parameters and results in increased accuracy of Monte Carlo estimates with little extra computational cost. S ince its introduction in the 1970s, the Metropolis−Hastings algorithm has revolutionized computational statistics (1). The ability to draw samples from an arbitrary probability distribution, πðXÞ, known only up to a constant, by constructing a Markov chain that converges to the correct stationary distribution has enabled the practical application of Bayesian inference for modeling a huge variety of scientific phenomena, and has resulted in Metropolis−Hastings being noted as one of the top 10 most important algorithms from the 20th century (2). Despite regular increases in available computing power, Markov chain Monte Carlo (MCMC) algorithms can still be computationally very expensive; many thousands of iterations may be necessary to obtain low-variance estimates of the required quantities with an oftentimes complex statistical model being evaluated for each set of proposed parameters. Furthermore, many Metropolis−Hastings algorithms are severely limited by their inherently sequential nature.Many approaches have been proposed for improving the statistical efficiency of MCMC, and although such algorithms are guaranteed to converge asymptotically to the stationary distribution, their performance over a finite number of iterations can vary hugely. Much research effort has therefore focused on developing transition kernels that enable moves to be proposed far from the current point and subsequently accepted with high probability, taking into account, for example, the correlation structure of the parameter space (3, 4), or by using Hamiltonian dynamics (5) or diffusion processes (6). A recent investigation into proposal kernels suggests that more exotic distributions, such as the Bactrian kernel, might also be used to increase the statistical efficiency of MCMC algorithms (7). The efficient exploration of high-dimensional and multimo...

show abstract

“…This not only makes it difficult to rapidly explore multidimensional parameter spaces, but also complicates assessment of convergence, and effective use of multiprocessor resources. Prefetching [Brockwell, 2006] and multitry Metropolis sampling [Liu et al, 2000] offer some options for distributed, multicore implementation of single chains. What is more, the use of a single chain increases chances of premature convergence.…”

Section: Comparison With Other Posterior Sampling Methods 331 Compmentioning

confidence: 99%

Probabilistic inference of multi‐Gaussian fields from indirect hydrological data using circulant embedding and dimensionality reduction

Laloy

Linde

Jacques

et al. 2015

Water Resources Research

View full text Add to dashboard Cite

We present a Bayesian inversion method for the joint inference of high-dimensional multiGaussian hydraulic conductivity fields and associated geostatistical parameters from indirect hydrological data. We combine Gaussian process generation via circulant embedding to decouple the variogram from grid cell specific values, with dimensionality reduction by interpolation to enable Markov chain Monte Carlo (MCMC) simulation. Using the Matern variogram model, this formulation allows inferring the conductivity values simultaneously with the field smoothness (also called Matern shape parameter) and other geostatistical parameters such as the mean, sill, integral scales and anisotropy direction(s) and ratio(s). The proposed dimensionality reduction method systematically honors the underlying variogram and is demonstrated to achieve better performance than the Karhunen-Loè ve expansion. We illustrate our inversion approach using synthetic (error corrupted) data from a tracer experiment in a fairly heterogeneous 10,000-dimensional 2-D conductivity field. A 40-times reduction of the size of the parameter space did not prevent the posterior simulations to appropriately fit the measurement data and the posterior parameter distributions to include the true geostatistical parameter values. Overall, the posterior field realizations covered a wide range of geostatistical models, questioning the common practice of assuming a fixed variogram prior to inference of the hydraulic conductivity values. Our method is shown to be more efficient than sequential Gibbs sampling (SGS) for the considered case study, particularly when implemented on a distributed computing cluster. It is also found to outperform the method of anchored distributions (MAD) for the same computational budget.

show abstract

Parallel Markov chain Monte Carlo Simulation by Pre-Fetching

Cited by 96 publications

References 18 publications

Dielectric Properties of Water Ice, the Ice Ih/XI Phase Transition, and an Assessment of Density Functional Theory

Dielectric Properties of Water Ice, the Ice Ih/XI Phase Transition, and an Assessment of Density Functional Theory

A general construction for parallelizing Metropolis−Hastings algorithms

Probabilistic inference of multi‐Gaussian fields from indirect hydrological data using circulant embedding and dimensionality reduction

Contact Info

Product

Resources

About