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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–Loéve expansion for square-integrable random variables can be used to sample such measures on quasi-Banach spaces. 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.

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