TY - CHAP A1 - Sullivan, T. J. T1 - Well-posedness of Bayesian inverse problems in quasi-Banach spaces with stable priors T2 - 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Weimar 2017 Y1 - 2017 U6 - https://doi.org/10.1002/pamm.201710402 VL - 17 IS - 1 SP - 871 EP - 874 ER - TY - JOUR A1 - Sullivan, T. J. T1 - Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors JF - Inverse Problems and Imaging Y1 - 2017 U6 - https://doi.org/10.3934/ipi.2017040 VL - 11 IS - 5 SP - 857 EP - 874 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 - TY - JOUR A1 - Klebanov, Ilja A1 - Sprungk, Björn A1 - Sullivan, T. J. T1 - The linear conditional expectation in Hilbert space JF - Bernoulli Y1 - 2021 U6 - https://doi.org/10.3150/20-BEJ1308 VL - 27 IS - 4 SP - 2299 EP - 2299 ER - TY - JOUR A1 - Lie, Han Cheng A1 - Sullivan, T. J. A1 - Stuart, Andrew T1 - Strong convergence rates of probabilistic integrators for ordinary differential equations JF - Statistics and Computing N2 - Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors inroduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\ Comput.}, 2016), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. Y1 - 2019 U6 - https://doi.org/10.1007/s11222-019-09898-6 VL - 29 IS - 6 SP - 1265 EP - 1283 ER - 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 - JOUR A1 - Lie, Han Cheng A1 - Sullivan, T. J. T1 - Quasi-invariance of countable products of Cauchy measures under non-unitary dilations JF - Electronic Communications in Probability Y1 - 2018 U6 - https://doi.org/10.1214/18-ECP113 VL - 23 IS - 8 SP - 1 EP - 6 ER - TY - CHAP A1 - Cockayne, Jon A1 - Oates, Chris A1 - Sullivan, T. J. A1 - Girolami, Mark T1 - Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems T2 - Proceedings of the 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering N2 - This paper develops meshless methods for probabilistically describing discretisation error in the numerical solution of partial differential equations. This construction enables the solution of Bayesian inverse problems while accounting for the impact of the discretisation of the forward problem. In particular, this drives statistical inferences to be more conservative in the presence of significant solver error. Theoretical results are presented describing rates of convergence for the posteriors in both the forward and inverse problems. This method is tested on a challenging inverse problem with a nonlinear forward model Y1 - 2017 U6 - https://doi.org/10.1063/1.4985359 VL - 1853 SP - 060001-1 EP - 060001-8 ER - TY - GEN A1 - Cockayne, Jon A1 - Oates, Chris A1 - Sullivan, T. J. A1 - Girolami, Mark T1 - Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems N2 - This paper develops a class of meshless methods that are well-suited to statistical inverse problems involving partial differential equations (PDEs). The methods discussed in this paper view the forcing term in the PDE as a random field that induces a probability distribution over the residual error of a symmetric collocation method. This construction enables the solution of challenging inverse problems while accounting, in a rigorous way, for the impact of the discretisation of the forward problem. In particular, this confers robustness to failure of meshless methods, with statistical inferences driven to be more conservative in the presence of significant solver error. In addition, (i) a principled learning-theoretic approach to minimise the impact of solver error is developed, and (ii) the challenging setting of inverse problems with a non-linear forward model is considered. The method is applied to parameter inference problems in which non-negligible solver error must be accounted for in order to draw valid statistical conclusions. T3 - ZIB-Report - 16-31 KW - Probabilistic Numerics KW - Partial Differential Equations KW - Inverse Problems KW - Meshless Methods KW - Gaussian Processes KW - Pseudo-Marginal MCMC Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-59513 SN - 1438-0064 ER - TY - GEN A1 - Oates, Chris A1 - Cockayne, Jon A1 - Prangle, Dennis A1 - Sullivan, T. J. A1 - Girolami, Mark ED - Hickernell, F. J. ED - Kritzer, P. T1 - Optimality criteria for probabilistic numerical methods T2 - Multivariate Algorithms and Information-Based Complexity N2 - It is well understood that Bayesian decision theory and average case analysis are essentially identical. However, if one is interested in performing uncertainty quantification for a numerical task, it can be argued that the decision-theoretic framework is neither appropriate nor sufficient. To this end, we consider an alternative optimality criterion from Bayesian experimental design and study its implied optimal information in the numerical context. This information is demonstrated to differ, in general, from the information that would be used in an average-case-optimal numerical method. The explicit connection to Bayesian experimental design suggests several distinct regimes in which optimal probabilistic numerical methods can be developed. Y1 - 2020 U6 - https://doi.org/10.1515/9783110635461-005 VL - 27 SP - 65 EP - 88 PB - De Gruyter ER -