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

A revised version of “Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors” has been released on arXiv today. Among other improvements, the revised version incorporates additional remarks on the connection to the existing literature on stable distributions in Banach spaces, and generalises the results of the previous version of the paper to quasi-Banach spaces, which are like complete normed vector spaces in every respect except that the triangle inequality only holds in the weakened form

\( \| x + y \| \leq C ( \| x \| + \| y \| ) \)

for some constant \( C \geq 1 \).

Published on Monday 21 November 2016 at 11:30 UTC #publication #preprint #inverse-problems

Cameron-Martin theorems for Cauchy-distributed random sequences

Preprint: Cameron-Martin theorems for Cauchy-distributed random sequences

Han Cheng Lie and I have just uploaded a preprint of our latest paper, on Cameron–Martin-type theorems for sequences of Cauchy-distributed random variables, to the arXiv. Inspired by questions of prior robustness left unanswered in this earlier paper on α-stable Banach space priors, this paper addresses the basic probabilistic question: when is an infinite-dimensional Cauchy distribution, e.g. on sequence space, mutually absolutely continuous with its image under a translation? In the Gaussian case, the celebrated Cameron–Martin theorem says that this equivalence of measures holds if a weighted \(\ell^{2}\) norm (the Cameron–Martin norm) of the translation vector is finite. We show that, in the Cauchy case, the same weighted version of the translation vector needs to lie in the sequence space \(\ell^{1} \cap \ell \log \ell\). More precisely, if the Cauchy distribution on the nth term of the sequence has width parameter \(\gamma_{n} > 0\), and the translation vector is the sequence \(\varepsilon = (\varepsilon_n)_{n = 1}^{\infty}\), then a sufficient condition for mutual absolute continuity is that

\( \displaystyle \sum_{n = 1}^{\infty} \left| \frac{\varepsilon_{n}}{\gamma_{n}} \right| < \infty \)

and, with the usual convention that \(0 \log 0 = 0\),

\( \displaystyle \sum_{n = 1}^{\infty} \left| \frac{\varepsilon_{n}}{\gamma_{n}} \log \left| \frac{\varepsilon_{n}}{\gamma_{n}} \right| \right| < \infty . \)

We also discuss similar results for dilation of the scale parameters, i.e. \(\gamma_{n} \mapsto \sigma_{n} \gamma_{n}\) for some real sequence \(\sigma = (\sigma_n)_{n = 1}^{\infty}\).

Published on Monday 15 August 2016 at 10:00 UTC #publication #preprint

Probabilistic meshless methods for partial differential equations and Bayesian inverse problems

Preprint: Probabilistic meshless methods for PDEs and BIPs

Jon Cockayne, Chris Oates, Mark Girolami and I have just uploaded a preprint of our latest paper, “Probabilistic meshless methods for partial differential equations and Bayesian inverse problems” to the arXiv. This paper forms part of the push for probabilistic numerics in scientific computing.

Abstract. This paper develops a class of meshless methods that are well-suited to statistical inverse problems involving partial differential equations (PDEs). The methods discussed in this paper view the forcing term in the PDE as a random field that induces a probability distribution over the residual error of a symmetric collocation method. This construction enables the solution of challenging inverse problems while accounting, in a rigorous way, for the impact of the discretisation of the forward problem. In particular, this confers robustness to failure of meshless methods, with statistical inferences driven to be more conservative in the presence of significant solver error. In addition, (i) a principled learning-theoretic approach to minimise the impact of solver error is developed, and (ii) the challenging setting of inverse problems with a non-linear forward model is considered. The method is applied to parameter inference problems in which non-negligible solver error must be accounted for in order to draw valid statistical conclusions.

Published on Thursday 26 May 2016 at 09:00 UTC #publication #preprint #prob-num

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

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