A comparison of variational and Markov chain Monte Carlo methods for inference in partially observed stochastic dynamic systems

In recent work we have developed a novel variational inference method for partially observed systems governed by stochastic differential equations. In this paper we provide a comparison of the Variational Gaussian Process Smoother with an exact solution computed using a Hybrid Monte Carlo approach to path sampling, applied to a stochastic double well potential model. It is demonstrated that the variational smoother provides us a very accurate estimate of mean path while conditional variance is slightly underestimated. We conclude with some remarks as to the advantages and disadvantages of the variational smoother. © 2008 Springer Science + Business Media LLC.

Variational Markov Chain Monte Carlo for Bayesian smoothing of non-linear niffusions

In this paper we develop set of novel Markov chain Monte Carlo algorithms for Bayesian smoothing of partially observed non-linear diffusion processes. The sampling algorithms developed herein use a deterministic approximation to the posterior distribution over paths as the proposal distribution for a mixture of an independence and a random walk sampler. The approximating distribution is sampled by simulating an optimized time-dependent linear diffusion process derived from the recently developed variational Gaussian process approximation method. Flexible blocking strategies are introduced to further improve mixing, and thus the efficiency, of the sampling algorithms. The algorithms are tested on two diffusion processes: one with double-well potential drift and another with SINE drift. The new algorithm's accuracy and efficiency is compared with state-of-the-art hybrid Monte Carlo based path sampling. It is shown that in practical, finite sample, applications the algorithm is accurate except in the presence of large observation errors and low observation densities, which lead to a multi-modal structure in the posterior distribution over paths. More importantly, the variational approximation assisted sampling algorithm outperforms hybrid Monte Carlo in terms of computational efficiency, except when the diffusion process is densely observed with small errors in which case both algorithms are equally efficient.

Variational Markov chain Monte Carlo for Bayesian smoothing of non-linear diffusions

In this paper we develop set of novel Markov Chain Monte Carlo algorithms for Bayesian smoothing of partially observed non-linear diffusion processes. The sampling algorithms developed herein use a deterministic approximation to the posterior distribution over paths as the proposal distribution for a mixture of an independence and a random walk sampler. The approximating distribution is sampled by simulating an optimized time-dependent linear diffusion process derived from the recently developed variational Gaussian process approximation method. The novel diffusion bridge proposal derived from the variational approximation allows the use of a flexible blocking strategy that further improves mixing, and thus the efficiency, of the sampling algorithms. The algorithms are tested on two diffusion processes: one with double-well potential drift and another with SINE drift. The new algorithm's accuracy and efficiency is compared with state-of-the-art hybrid Monte Carlo based path sampling. It is shown that in practical, finite sample applications the algorithm is accurate except in the presence of large observation errors and low to a multi-modal structure in the posterior distribution over paths. More importantly, the variational approximation assisted sampling algorithm outperforms hybrid Monte Carlo in terms of computational efficiency, except when the diffusion process is densely observed with small errors in which case both algorithms are equally efficient. © 2011 Springer-Verlag.

Productivity change using growth accounting and frontier-based approaches:evidence from a Monte Carlo analysis

This study presents some quantitative evidence from a number of simulation experiments on the accuracy of the productivitygrowth estimates derived from growthaccounting (GA) and frontier-based methods (namely data envelopment analysis-, corrected ordinary least squares-, and stochastic frontier analysis-based malmquist indices) under various conditions. These include the presence of technical inefficiency, measurement error, misspecification of the production function (for the GA and parametric approaches) and increased input and price volatility from one period to the next. The study finds that the frontier-based methods usually outperform GA, but the overall performance varies by experiment. Parametric approaches generally perform best when there is no functional form misspecification, but their accuracy greatly diminishes otherwise. The results also show that the deterministic approaches perform adequately even under conditions of (modest) measurement error and when measurement error becomes larger, the accuracy of all approaches (including stochastic approaches) deteriorates rapidly, to the point that their estimates could be considered unreliable for policy purposes.

Monte Carlo and molecular dynamics simulations of methane in potassium montmorillonite clay hydrates at elevated pressures and temperatures

The structure and dynamics of methane in hydrated potassium montmorillonite clay have been studied under conditions encountered in sedimentary basin and compared to those of hydrated sodium montmorillonite clay using computer simulation techniques. The simulated systems contain two molecular layers of water and followed gradients of 150 barkm-1 and 30 Kkm-1 up to a maximum burial depth of 6 km. Methane particle is coordinated to about 19 oxygen atoms, with 6 of these coming from the clay surface oxygen. Potassium ions tend to move away from the center towards the clay surface, in contrast to the behavior observed with the hydrated sodium form. The clay surface affinity for methane was found to be higher in the hydrated K-form. Methane diffusion in the two-layer hydrated K-montmorillonite increases from 0.39×10-9 m2s-1 at 280 K to 3.27×10-9 m2s-1 at 460 K compared to 0.36×10-9 m2s-1 at 280 K to 4.26×10-9 m2s-1 at 460 K in Na-montmorillonite hydrate. The distributions of the potassium ions were found to vary in the hydrates when compared to those of sodium form. Water molecules were also found to be very mobile in the potassium clay hydrates compared to sodium clay hydrates. © 2004 Elsevier Inc. All All rights reserved.

Accuracy and optimal sampling in Monte Carlo solution of population balance equations

Implementation of a Monte Carlo simulation for the solution of population balance equations (PBEs) requires choice of initial sample number (N0), number of replicates (M), and number of bins for probability distribution reconstruction (n). It is found that Squared Hellinger Distance, H2, is a useful measurement of the accuracy of Monte Carlo (MC) simulation, and can be related directly to N0, M, and n. Asymptotic approximations of H2 are deduced and tested for both one-dimensional (1-D) and 2-D PBEs with coalescence. The central processing unit (CPU) cost, C, is found in a power-law relationship, C= aMNb0, with the CPU cost index, b, indicating the weighting of N0 in the total CPU cost. n must be chosen to balance accuracy and resolution. For fixed n, M × N0 determines the accuracy of MC prediction; if b > 1, then the optimal solution strategy uses multiple replications and small sample size. Conversely, if 0 < b < 1, one replicate and a large initial sample size is preferred. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2394–2402, 2015