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

摘要

This article extends the framework of Bayesian inverse problems ininfinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer.19:451--559, 2010) and others, to the case of a heavy-tailed prior measure inthe family of stable distributions, such as an infinite-dimensional Cauchydistribution, for which polynomial moments are infinite or undefined. It isshown that analogues of the Karhunen--Lo\`eve expansion for square-integrablerandom variables can be used to sample such measures on quasi-Banach spaces.Furthermore, under weaker regularity assumptions than those used to date, theBayesian posterior measure is shown to depend Lipschitz continuously in theHellinger metric upon perturbations of the misfit function and observed data.