Tim Sullivan

Junior Professor in Applied Mathematics:
Risk and Uncertainty Quantification

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

Preprint: Bayesian inversion with heavy-tailed stable priors

Just uploaded to the arXiv: “Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priors”. This article builds on the function-space formulation of Bayesian inverse problems advocated by Stuart et al. to allow the prior to be heavy-tailed: not only may it not be exponentially integrable, as is the case for a Gaussian or Besov measure, it might not even have a well-defined mean, as in the case of the famous Cauchy distribution on \(\mathbb{R}\).

Abstract. This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451–559, 2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen–Loève expansion for square-integrable random variables can be used to sample such measures. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger metric upon perturbations of the misfit function and observed data.

Published on Friday 20 May 2016 at 09:00 UTC #publication #preprint #inverse-problems

Introduction to Uncertainty Quantification

Errata for Introduction to Uncertainty Quantification

A list of errata, corrections, and clarifications for Introduction to Uncertainty Quantification can now be found here. In case you spot any mistakes that are not on this list, then please get in touch and I will be happy to post the correction on the errata page.

Published on Monday 11 April 2016 at 11:00 UTC #publication #i2uq

Introduction to Uncertainty Quantification

Introduction to Uncertainty Quantification Now Available as e-Book and Hardcover

A 350-page introduction to the key mathematical ideas underlying uncertainty quantification, designed as a course text or self-study for finalist undergraduates, master\'s students, or beginning doctoral students.

T. J. Sullivan. Introduction to Uncertainty Quantification, volume 63 of Texts in Applied Mathematics. Springer, 2015. ISBN 978-3-319-23394-9 (hardcover) 978-3-319-23395-6 (e-book) doi:10.1007/978-3-319-23395-6

Update, 11 March 2016. A list of errata can now be found here.

Published on Tuesday 22 December 2015 at 11:00 UTC #publication #i2uq

On the Brittleness of Bayesian Inference

Bayesian Brittleness in SIAM Review

The 2015 Q4 issue of SIAM Review will carry an article by Houman Owhadi, Clint Scovel, and myself on the brittle dependency of Bayesian posteriors as a function of the prior. This is an abbreviated presentation of results given in full earlier this year in Elec. J. Stat. The PDF is available for free under the terms of the Creative Commons 4.0 licence.

H. Owhadi, C. Scovel & T. J. Sullivan. “On the Brittleness of Bayesian Inference” SIAM Review 57(4):566–582, 2015. doi:10.1137/130938633

Published on Friday 6 November 2015 at 12:00 UTC #publication

Brittleness of Bayesian inference under finite information in a continuous world

Bayesian Brittleness in Elec. J. Stat.

The Electronic Journal of Statistics has published an article by Houman Owhadi, Clint Scovel, and myself on the brittle dependency of Bayesian posteriors as a function of the prior.

H. Owhadi, C. Scovel & T. J. Sullivan. “Brittleness of Bayesian inference under finite information in a continuous world” Electronic Journal of Statistics 9:1–79, 2015. doi:10.1214/15-EJS989

Published on Tuesday 3 February 2015 at 10:00 UTC #publication

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