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Bayesian probabilistic numerical methods

Bayesian probabilistic numerical methods in SIAM Review

The 2019 Q4 issue of SIAM Review will carry an article by Jon Cockayne, Chris Oates, Mark Girolami, and myself on the Bayesian formulation of probabilistic numerical methods, i.e. the interpretation of deterministic numerical tasks such as quadrature and the solution of ordinary and partial differential equations as (Bayesian) statistical inference tasks.

J. Cockayne, C. J. Oates, T. J. Sullivan, and M. Girolami. “Bayesian probabilistic numerical methods.” SIAM Review 61(4):756–789, 2019. doi:10.1137/17M1139357

Abstract. Over forty years ago average-case error was proposed in the applied mathematics literature as an alternative criterion with which to assess numerical methods. In contrast to worst-case error, this criterion relies on the construction of a probability measure over candidate numerical tasks, and numerical methods are assessed based on their average performance over those tasks with respect to the measure. This paper goes further and establishes Bayesian probabilistic numerical methods as solutions to certain inverse problems based upon the numerical task within the Bayesian framework. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well defined, encompassing both the nonlinear and non-Gaussian contexts. For general computation, a numerical approximation scheme is proposed and its asymptotic convergence established. The theoretical development is extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, and a challenging industrial application is presented.

Published on Thursday 7 November 2019 at 07:00 UTC #publication #bayesian #siam-review #prob-num #cockayne #girolami #oates

Random forward models and log-likelihoods in Bayesian inverse problems

Random Bayesian inverse problems in JUQ

The article “Random forward models and log-likelihoods in Bayesian inverse problems” by Han Cheng Lie, Aretha Teckentrup, and myself has now appeared in its final form in the SIAM/ASA Journal on Uncertainty Quantification, volume 6, issue 4. This paper considers the effect of approximating the likelihood in a Bayesian inverse problem by a random surrogate, as frequently happens in applications, with the aim of showing that the perturbed posterior distribution is close to the exact one in a suitable sense. This article considers general randomisation models, and thereby expands upon the previous investigations of Stuart and Teckentrup (2018) in the Gaussian setting.

H. C. Lie, T. J. Sullivan, and A. L. Teckentrup. “Random forward models and log-likelihoods in Bayesian inverse problems.” SIAM/ASA Journal on Uncertainty Quantification 6(4):1600–1629, 2018. doi:10.1137/18M1166523

Abstract. We consider the use of randomised forward models and log-likelihoods within the Bayesian approach to inverse problems. Such random approximations to the exact forward model or log-likelihood arise naturally when a computationally expensive model is approximated using a cheaper stochastic surrogate, as in Gaussian process emulation (kriging), or in the field of probabilistic numerical methods. We show that the Hellinger distance between the exact and approximate Bayesian posteriors is bounded by moments of the difference between the true and approximate log-likelihoods. Example applications of these stability results are given for randomised misfit models in large data applications and the probabilistic solution of ordinary differential equations.

Published on Monday 10 December 2018 at 12:00 UTC #publication #bayesian #inverse-problems #juq #prob-num #lie #teckentrup

Random forward models and log-likelihoods in Bayesian inverse problems

Preprint: Random Bayesian inverse problems

Han Cheng Lie, Aretha Teckentrup, and I have just uploaded a preprint of our latest article, “Random forward models and log-likelihoods in Bayesian inverse problems”, to the arXiv. This paper considers the effect of approximating the likelihood in a Bayesian inverse problem by a random surrogate, as frequently happens in applications, with the aim of showing that the perturbed posterior distribution is close to the exact one in a suitable sense. This article considers general randomisation models, and thereby expands upon the previous investigations of Stuart and Teckentrup (2018) in the Gaussian setting.

Abstract. We consider the use of randomised forward models and log-likelihoods within the Bayesian approach to inverse problems. Such random approximations to the exact forward model or log-likelihood arise naturally when a computationally expensive model is approximated using a cheaper stochastic surrogate, as in Gaussian process emulation (kriging), or in the field of probabilistic numerical methods. We show that the Hellinger distance between the exact and approximate Bayesian posteriors is bounded by moments of the difference between the true and approximate log-likelihoods. Example applications of these stability results are given for randomised misfit models in large data applications and the probabilistic solution of ordinary differential equations.

Published on Tuesday 19 December 2017 at 08:30 UTC #publication #preprint #bayesian #inverse-problems #lie #teckentrup

Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors

Heavy-tailed stable priors in Inverse Problems and Imaging

The final version of “Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors” has now been published online in Inverse Problems and Imaging; the print version will be available in October.

T. J. Sullivan. “Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors.” Inverse Problems and Imaging 11(5):857–874, 2017. doi:10.3934/ipi.2017040

Published on Wednesday 19 July 2017 at 15:30 UTC #publication #bayesian #inverse-problems #cauchy-distribution #inverse-problems-and-imaging

Bayesian probabilistic numerical methods

Preprint: Bayesian probabilistic numerical methods

Jon Cockayne, Chris Oates, Mark Girolami and I have just uploaded a preprint of our latest paper, “Bayesian probabilistic numerical methods” to the arXiv. Following on from our earlier work “Probabilistic meshless methods for partial differential equations and Bayesian inverse problems”, our aim is to provide some rigorous theoretical underpinnings for the emerging field of probabilistic numerics, and in particular to define what it means for such a method to be “Bayesian”, by connecting with the established theories of Bayesian inversion and disintegration of measures.

Abstract. The emergent field of probabilistic numerics has thus far lacked rigorous statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain Bayesian inverse problems, albeit problems that are non-standard. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well-defined, encompassing both non-linear and non-Gaussian models. For general computation, a numerical approximation scheme is developed and its asymptotic convergence is established. The theoretical development is then extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, with some illustrative applications presented.

Published on Tuesday 14 February 2017 at 12:00 UTC #publication #preprint #prob-num #bayesian #cockayne #oates #girolami