# Tim Sullivan

### Dilations of Cauchy measures in Electron. Commun. Prob.

“Quasi-invariance of countable products of Cauchy measures under non-unitary dilations”, by Han Cheng Lie and myself, has just appeared online in Electronic Communications in Probability. This main result of this article can be understood as an analogue of the celebrated Cameron–Martin theorem, which characterises the directions in which an infinite-dimensional Gaussian measure can be translated while preserving equivalence of the original and translated measure; our result is a similar characterisation of equivalence of measures, but for infinite-dimensional Cauchy measures under dilations instead of translations.

H. C. Lie & T. J. Sullivan. “Quasi-invariance of countable products of Cauchy measures under non-unitary dilations.” Electronic Communications in Probability 23(8):1–6, 2018. doi:10.1214/18-ECP113

Abstract. Consider an infinite sequence $$(U_{n})_{n \in \mathbb{N}}$$ of independent Cauchy random variables, defined by a sequence $$(\delta_{n})_{n \in \mathbb{N}}$$ of location parameters and a sequence $$(\gamma_{n})_{n \in \mathbb{N}}$$ of scale parameters. Let $$(W_{n})_{n \in \mathbb{N}}$$ be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence $$(\sigma_{n} \gamma_{n})_{n \in \mathbb{N}}$$ of scale parameters, with $$\sigma_{n} \neq 0$$ for all $$n \in \mathbb{N}$$. Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of $$(U_{n})_{n \in \mathbb{N}}$$ and $$(W_{n})_{n \in \mathbb{N}}$$ are equivalent if and only if the sequence $$(| \sigma_{n}| - 1 )_{n \in \mathbb{N}}$$ is square-summable.

Published on Wednesday 21 February 2018 at 09:30 UTC #publication #electron-commun-prob #cauchy-distribution

### Preprint: Random Bayesian inverse problems

Han Cheng Lie, Aretha Teckentrup, and I have just 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 (2017) 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 #inverse-problems

### Preprint: Active subspace Metropolis-Hastings

Ingmar Schuster, Paul Constantine and I have just uploaded a preprint of our latest article, “Exact active subspace Metropolis–Hastings, with applications to the Lorenz-96 system”, to the arXiv. This paper reports on our first investigations into the acceleration of Markov chain Monte Carlo methods using active subspaces as compared to other adaptivity techniques, and is supported by the DFG through SFB 1114 Scaling Cascades in Complex Systems.

Abstract. We consider the application of active subspaces to inform a Metropolis–Hastings algorithm, thereby aggressively reducing the computational dimension of the sampling problem. We show that the original formulation, as proposed by Constantine, Kent, and Bui-Thanh (SIAM J. Sci. Comput., 38(5):A2779–A2805, 2016), possesses asymptotic bias. Using pseudo-marginal arguments, we develop an asymptotically unbiased variant. Our algorithm is applied to a synthetic multimodal target distribution as well as a Bayesian formulation of a parameter inference problem for a Lorenz-96 system.

Published on Friday 8 December 2017 at 08:00 UTC #publication #preprint #mcmc #sfb1114

### Preprint: Weak and strong modes

Han Cheng Lie and I have just uploaded a preprint of our latest paper, “Equivalence of weak and strong modes of measures on topological vector spaces”, to the arXiv. This addresses a natural question in the theory of modes (or maximum a posteriori estimators, in the case of posterior measure for a Bayesian inverse problem) in infinite-dimensional spaces, which are defined either strongly (a la Dashti et al. (2013), via a global maximisation) or weakly (a la Helin & Burger (2015), via a dense subspace): when are strong and weak modes equivalent?

Abstract. Modes of a probability measure on an infinite-dimensional Banach space $$X$$ are often defined by maximising the small-radius limit of the ratio of measures of norm balls. Helin and Burger weakened the definition of such modes by considering only balls with centres in proper subspaces of $$X$$, and posed the question of when this restricted notion coincides with the unrestricted one. We generalise these definitions to modes of arbitrary measures on topological vector spaces, defined by arbitrary bounded, convex, neighbourhoods of the origin. We show that a coincident limiting ratios condition is a necessary and sufficient condition for the equivalence of these two types of modes, and show that the coincident limiting ratios condition is satisfied in a wide range of topological vector spaces.

Published on Wednesday 9 August 2017 at 05:00 UTC #publication #preprint #inverse-problems

### 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.

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

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