TY - GEN A1 - Lie, Han Cheng A1 - Sullivan, T. J. A1 - Teckentrup, Aretha T1 - Random forward models and log-likelihoods in Bayesian inverse problems T2 - SIAM/ASA Journal on Uncertainty Quantification N2 - We consider the use of randomised forward models and log-likelihoods within the Bayesian approach to inverse problems. Such random approximations to the exact forward model or log-likelihood arise naturally when a computationally expensive model is approximated using a cheaper stochastic surrogate, as in Gaussian process emulation (kriging), or in the field of probabilistic numerical methods. We show that the Hellinger distance between the exact and approximate Bayesian posteriors is bounded by moments of the difference between the true and approximate log-likelihoods. Example applications of these stability results are given for randomised misfit models in large data applications and the probabilistic solution of ordinary differential equations. T3 - ZIB-Report - 18-03 KW - Bayesian inverse problem KW - random likelihood KW - surrogate model KW - posterior consistency KW - probabilistic numerics KW - uncertainty quantification KW - randomised misfit Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-66324 SN - 1438-0064 VL - 6 IS - 4 SP - 1600 EP - 1629 ER - TY - GEN A1 - Sullivan, T. J. T1 - Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priors N2 - 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. T3 - ZIB-Report - 16-30 KW - Bayesian inverse problems KW - heavy-tailed distribution KW - Karhunen–Loève expansion KW - stable distribution KW - uncertainty quantification KW - well-posedness Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-59422 SN - 1438-0064 ER -