Markov chain Monte Carlo methods: an introductory example

Klauenberg, Katy; Elster, Clemens

doi:10.1088/0026-1394/53/1/s32

Cited by 26 publications

(19 citation statements)

References 18 publications

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“…The surrogate can thus be exploited to perform a Markow Chain Monte Carlo (MCMC) sampling of the posterior density. 31,32 Performing such a sampling of the actual objective function is computationally very expensive and can take many days. 33 The use of an accurate surrogate model has thus the potential to accelerate MCMC analyses considerably.…”

Section: Discussionmentioning

confidence: 99%

Recent advances in Bayesian optimization with applications to parameter reconstruction in optical nano-metrology

Plock,

Burger,

Schneider

2021

Preprint

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Parameter reconstruction is a common problem in optical nano metrology. It generally involves a set of measurements, to which one attempts to fit a numerical model of the measurement process. The model evaluation typically involves to solve Maxwell's equations and is thus time consuming. This makes the reconstruction computationally demanding. Several methods exist for fitting the model to the measurements. On the one hand, Bayesian optimization methods for expensive black-box optimization enable an efficient reconstruction by training a machine learning model of the squared sum of deviations χ 2 . On the other hand, curve fitting algorithms, such as the Levenberg-Marquardt method, take the deviations between all model outputs and corresponding measurement values into account which enables a fast local convergence. In this paper we present a Bayesian Target Vector Optimization scheme which combines these two approaches. We compare the performance of the presented method against a standard Levenberg-Marquardt-like algorithm, a conventional Bayesian optimization scheme, and the L-BFGS-B and Nelder-Mead simplex algorithms. As a stand-in for problems from nano metrology, we employ a non-linear least-square problem from the NIST Standard Reference Database. We find that the presented method generally uses fewer calls of the model function than any of the competing schemes to achieve similar reconstruction performance.

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Section: Discussionmentioning

confidence: 99%

Recent advances in Bayesian optimization with applications to parameter reconstruction in optical nano-metrology

Plock,

Burger,

Schneider

2021

Preprint

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“…The implementation of the corresponding probability calculus can then be employed by using Markov chain Monte Carlo methods, cf. [23,24] or approximation methods [25]. In order to take advantage of the sequential character of the filtering process, typically sequential methods are advocated.…”

Section: Particle Filtering and Related Methodsmentioning

confidence: 99%

On the evaluation of uncertainties for state estimation with the Kalman filter

Eichstädt

Makarava

Elster

2016

Meas. Sci. Technol.

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The Kalman filter is an established tool for the analysis of dynamic systems with normally distributed noise, and it has been successfully applied in numerous application areas. It provides sequentially calculated estimates of the system states along with a corresponding covariance matrix. For nonlinear systems, the extended Kalman filter is often used which is derived from the Kalman filter by linearization around the current estimate. A key issue in metrology is the evaluation of the uncertainty associated with the Kalman filter state estimates. The "Guide to the Expression of Uncertainty in Measurements" (GUM) and its supplements serve as the de facto standard for uncertainty evaluation in metrology. We explore the relationship between the covariance matrix produced by the Kalman filter and a GUM-compliant uncertainty analysis. In addition, also the results of a Bayesian analysis are considered. For the case of linear systems with known system matrices, we show that all three approaches are compatible. When the system matrices are not precisely known, however, or when the system is nonlinear, this equivalence breaks down and different results can be reached then. Though for precisely known nonlinear systems the result of the extended Kalman filter still corresponds to the linearized uncertainty propagation of GUM. The extended Kalman filter can suffer from linearization and convergence errors. These disadvantages can be avoided to some extent by applying Monte Carlo procedures, and we propose such a method which is GUMcompliant and can also be applied online during the estimation. We illustrate all procedures in terms of a two-dimensional dynamic system and compare the results with those obtained by particle filtering, which has been proposed for the approximate calculation of a Bayesian solution. Finally, we give some recommendations based on our findings.

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“…The Markov Chain Monte Carlo (MCMC) method is helpful for numerical estimation of ( | ) by formula (5), since it allows us to skip the normalizing constant factor ( ) [12,15].…”

Section: Accepted Manuscriptmentioning

confidence: 99%

“…At each iteration, a ratio equivalent to that in equation ( 10) is estimated and a new value from uniform distribution on (0,1) is drawn to decide which value to retain for the next iteration. This sampling method of the posterior distribution is known as the 'Metropolis-Hastings algorithm' [12,15].…”

Section: Accepted Manuscriptmentioning

confidence: 99%

“…If the starting point of the Markov Chain process is far away from the mode of the posterior distribution, it is convenient to reject the first iterations since they are not representative. The so-called "burn-in period" set for the Markov Chain defines the number of initial rejected iterations for the characterisation of the posterior distribution [12,15].…”

Section: Accepted Manuscriptmentioning

confidence: 99%

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