On the convergence of the Laplace approximation and noise-level-robustness of Laplace-based Monte Carlo methods for Bayesian inverse problems

Schillings, Claudia; Sprungk, Björn; Wacker, Philipp

doi:10.1007/s00211-020-01131-1

Cited by 64 publications

(55 citation statements)

References 42 publications

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“…In case of the current version of the BayesFactor package, such an option is an alternative algorithm using Laplace approximation. Because this optional algorithm does not use any sampling, it will provide stable results across runs (Schillings, Sprungk, & Wacker, 2020; for alternative algorithms with stable, analytic results, see Chen, Villa, & Ghattas, 2017;Schillings & Schwab, 2013). At the same time, Laplace approximation can return systematically different results than (many iterations of) MCMC-based methods, especially when sample sizes are small.…”

Section: Discussionmentioning

confidence: 99%

Variability of Bayes Factor estimates in Bayesian Analysis of Variance

Pfister¹

2021

TQMP

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Bayes Factor estimation for Bayesian Analysis of Variance (ANOVA) typically relies on iterative algorithms that, by design, yield slightly different results on every run of the analysis. The variability of these estimates is surprisingly large, however: The present simulations indicate that repeating one and the same Bayesian ANOVA on a constant dataset often results in Bayes Factors that differ by a factor of 2 or more within only a few runs when using common analysis procedures. Results may at times even suggest evidence for the null hypothesis of no effect on one run while supporting the alternative hypothesis on another run. These observations call for a cautious approach to the results of Bayesian ANOVAs at present, and I outline three possibilities to circumvent or minimize this limitation.

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Section: Discussionmentioning

confidence: 99%

Variability of Bayes Factor estimates in Bayesian Analysis of Variance

Pfister¹

2021

TQMP

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“…The Laplace approximation is exact when the parameter-to-observable map f(m) is linear, and often provides a reasonable approximation to the posterior, since the error is of the order of the departure from linearity of the parameter-to-observable map (Helin and Kretschmann 2020). Moreover, in the limit of small noise or large data, the Laplace approximation converges to the posterior (Schillings, Sprungk and Wacker 2020).…”

Section: Bayesian Formulationmentioning

confidence: 99%

Learning physics-based models from data: perspectives from inverse problems and model reduction

Ghattas¹,

Willcox²

2021

Acta Numerica

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This article addresses the inference of physics models from data, from the perspectives of inverse problems and model reduction. These fields develop formulations that integrate data into physics-based models while exploiting the fact that many mathematical models of natural and engineered systems exhibit an intrinsically low-dimensional solution manifold. In inverse problems, we seek to infer uncertain components of the inputs from observations of the outputs, while in model reduction we seek low-dimensional models that explicitly capture the salient features of the input–output map through approximation in a low-dimensional subspace. In both cases, the result is a predictive model that reflects data-driven learning yet deeply embeds the underlying physics, and thus can be used for design, control and decision-making, often with quantified uncertainties. We highlight recent developments in scalable and efficient algorithms for inverse problems and model reduction governed by large-scale models in the form of partial differential equations. Several illustrative applications to large-scale complex problems across different domains of science and engineering are provided.

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“…Additional tricks are necessary to make such methods work for very concentrated posterior distributions. In Schillings et al (2019), estimators of this type are derived with performance that is independent of the noise level in the observations. It would be interesting to investigate their applicability in a MLMC context.…”

Section: Approaches For Multilevel Monte Carlo Estimationmentioning

confidence: 99%

Advanced Multilevel Monte Carlo Methods

Jasra

Law

Suciu

2020

Int Statistical Rev

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This article reviews the application of advanced Monte Carlo techniques in the context of Multilevel Monte Carlo (MLMC). MLMC is a strategy employed to compute expectations which can be biased in some sense, for instance, by using the discretization of a associated probability law. The MLMC approach works with a hierarchy of biased approximations which become progressively more accurate and more expensive. Using a telescoping representation of the most accurate approximation, the method is able to reduce the computational cost for a given level of error versus i.i.d. sampling from this latter approximation. All of these ideas originated for cases where exact sampling from couples in the hierarchy is possible. This article considers the case where such exact sampling is not currently possible. We consider Markov chain Monte Carlo and sequential Monte Carlo methods which have been introduced in the literature and we describe different strategies which facilitate the application of MLMC within these methods.

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On the convergence of the Laplace approximation and noise-level-robustness of Laplace-based Monte Carlo methods for Bayesian inverse problems

Cited by 64 publications

References 42 publications

Variability of Bayes Factor estimates in Bayesian Analysis of Variance

Variability of Bayes Factor estimates in Bayesian Analysis of Variance

Learning physics-based models from data: perspectives from inverse problems and model reduction

Advanced Multilevel Monte Carlo Methods

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