Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors
Provides theoretical foundations for using heavy-tailed priors in Bayesian inverse problems, enabling robust inference in settings where standard Gaussian priors are inadequate.
The paper extends Bayesian inverse problems to heavy-tailed stable priors (e.g., Cauchy) with infinite moments, proving Lipschitz continuity of the posterior in Hellinger metric under weaker assumptions.
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 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.