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The Bayesian perspective on inverse problems has attracted much mathematicalattention in recent years. Particular attention has been paid to Bayesianinverse problems (BIPs) in which the parameter to be inferred lies in aninfinite-dimensional space, a typical example being a scalar or tensor fieldcoupled to some observed data via an ODE or PDE. This article gives anintroduction to the framework of well-posed BIPs in infinite-dimensionalparameter spaces, as advocated by Stuart (Acta Numer. 19:451--559, 2010) andothers. This framework has the advantage of ensuring uniformly well-posedinference problems independently of the finite-dimensional discretisation usedfor numerical solution. Recently, this framework has been extended to the caseof a heavy-tailed prior measure in the family of stable distributions, such asan infinite-dimensional Cauchy distribution, for which polynomial moments areinfinite or undefined. It is shown that analogues of the Karhunen--Lo\`eveexpansion for square-integrable random variables can be used to sample suchmeasures on quasi-Banach spaces. Furthermore, under weaker regularityassumptions than those used to date, the Bayesian posterior measure is shown todepend Lipschitz continuously in the Hellinger and total variation metrics uponperturbations of the misfit function and observed data.

Well-posedness of Bayesian inverse problems in quasi-Banach spaces with stable priors