Convergence Diagnostics for Markov Chain Monte Carlo

Roy, Vivekananda

doi:10.1146/annurev-statistics-031219-041300

Cited by 205 publications

(124 citation statements)

References 78 publications

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“…To understand why HMC works, we refer readers to the approachable and intuitive expositions in [22] and [24,Cha.15] for expert explanations of the algorithm and to [23], [25]- [27] for further technical analysis. In particular, Betancourt discusses how HMC is "uniquely suited to the high-dimensional problems of applied interest."…”

Section: Algorithm 1 Hamiltonian Monte Carlomentioning

confidence: 99%

“…The No-U-Turn Sampler has at least the same efficiency as a well-tuned HMC algorithm [25]. The convergence is usually checked by empirical diagnostic tools [27]. Also, we carefully set the initial values of the parameters to make the convergence faster by exploring the outage data in Section III.…”

Section: Algorithm 1 Hamiltonian Monte Carlomentioning

confidence: 99%

“…The Gelman-Rubin potential scale reduction factor diag-nosticR is often used to check Markov chain Monte Carlo convergence [27].R is defined as the ratio of the estimated pooled variance to the estimated within-chain variance (see [8,Sec. 11.4] for the equations ofR).…”

Section: Appendix B Convergence Of Sampling Algorithmmentioning

confidence: 99%

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Bayesian Estimates of Transmission Line Outage Rates That Consider Line Dependencies

Zhou

Cruise

Dent

et al. 2021

IEEE Trans. Power Syst.

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Transmission line outage rates are fundamental to power system reliability analysis. Line outages are infrequent, occurring only about once a year, so outage data are limited. We propose a Bayesian hierarchical model that leverages line dependencies to better estimate outage rates of individual transmission lines from limited outage data. The Bayesian estimates have a lower standard deviation than estimating the outage rates simply by dividing the number of outages by the number of years of data, especially when the number of outages is small. The Bayesian model produces more accurate individual line outage rates, as well as estimates of the uncertainty of these rates. Better estimates of line outage rates can improve system risk assessment, outage prediction, and maintenance scheduling.

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Section: Algorithm 1 Hamiltonian Monte Carlomentioning

confidence: 99%

Section: Algorithm 1 Hamiltonian Monte Carlomentioning

confidence: 99%

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Bayesian Estimates of Transmission Line Outage Rates That Consider Line Dependencies

Zhou

Cruise

Dent

et al. 2021

IEEE Trans. Power Syst.

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“…For HMC-NUTS, we run 100 times for the dual averaging algorithm to tune the Störmer-Verlet symplectic integrator, and then we generate 4,000 samples for uncertainty quantification. For MCMC convergence analysis, we use Geweke z-score test for measuring stationarity of the chain and autocorrelation function and time τ for checking chain mixing (Geweke, 1992;Roy, 2020). We also convert the z-score into a two-sided p-value, for certain significance level comparisons.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Enhancing industrial X-ray tomography by data-centric statistical methods

Suuronen

Emzir

Lasanen

et al. 2020

DCE

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X-ray tomography has applications in various industrial fields such as sawmill industry, oil and gas industry, as well as chemical, biomedical, and geotechnical engineering. In this article, we study Bayesian methods for the X-ray tomography reconstruction. In Bayesian methods, the inverse problem of tomographic reconstruction is solved with the help of a statistical prior distribution which encodes the possible internal structures by assigning probabilities for smoothness and edge distribution of the object. We compare Gaussian random field priors, that favor smoothness, to non-Gaussian total variation (TV), Besov, and Cauchy priors which promote sharp edges and high- and low-contrast areas in the object. We also present computational schemes for solving the resulting high-dimensional Bayesian inverse problem with 100,000–1,000,000 unknowns. We study the applicability of a no-U-turn variant of Hamiltonian Monte Carlo (HMC) methods and of a more classical adaptive Metropolis-within-Gibbs (MwG) algorithm to enable full uncertainty quantification of the reconstructions. We use maximum a posteriori (MAP) estimates with limited-memory BFGS (Broyden–Fletcher–Goldfarb–Shanno) optimization algorithm. As the first industrial application, we consider sawmill industry X-ray log tomography. The logs have knots, rotten parts, and even possibly metallic pieces, making them good examples for non-Gaussian priors. Secondly, we study drill-core rock sample tomography, an example from oil and gas industry. In that case, we compare the priors without uncertainty quantification. We show that Cauchy priors produce smaller number of artefacts than other choices, especially with sparse high-noise measurements, and choosing HMC enables systematic uncertainty quantification, provided that the posterior is not pathologically multimodal or heavy-tailed.

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“…At best, we hope to “fail to prove a failure to converge”. A plethora of convergence criteria and/or graphical methods is available to do so 24 . For example, we may create a trace plot showing the values of the parameter generated at each iteration and check whether they converge to stationary values.…”

Section: Convergencementioning

confidence: 99%

Study design synopsis: Battle in the stable: Bayesianism versus Frequentism

Detilleux¹

2020

Equine Veterinary Journal

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Frequentism dominates scientific practice although Bayesianism may provide an alternative, especially when analysing data from complex, high‐dimensional models. The key differences between Bayesianism and Frequentism are highlighted in the introduction. Next, I review the different stages of Bayesian statistical reasoning in a research setting, explain the key concepts and illustrate them with toy examples taken in equine veterinary medicine. An extension to more complex models (Bayes network) is introduced and guidelines are offered as a conclusion.

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Convergence Diagnostics for Markov Chain Monte Carlo

Cited by 205 publications

References 78 publications

Bayesian Estimates of Transmission Line Outage Rates That Consider Line Dependencies

Bayesian Estimates of Transmission Line Outage Rates That Consider Line Dependencies

Enhancing industrial X-ray tomography by data-centric statistical methods

Study design synopsis: Battle in the stable: Bayesianism versus Frequentism

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