# Tim Sullivan

Junior Professor in Applied Mathematics:
Risk and Uncertainty Quantification

### 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