Dokument-ID Dokumenttyp Verfasser/Autoren Herausgeber Haupttitel Abstract Auflage Verlagsort Verlag Erscheinungsjahr Seitenzahl Schriftenreihe Titel Schriftenreihe Bandzahl ISBN Quelle der Hochschulschrift Konferenzname Quelle:Titel Quelle:Jahrgang Quelle:Heftnummer Quelle:Erste Seite Quelle:Letzte Seite URN DOI Abteilungen
OPUS4-6023 misc Lie, Han Cheng; Sullivan, T. J. Cameron--Martin theorems for sequences of Cauchy-distributed random variables Given a sequence of Cauchy-distributed random variables defined by a sequence of location parameters and a sequence of scale parameters, we consider another sequence of random variables that is obtained by perturbing the location or scale parameter sequences. Using a result of Kakutani on equivalence of infinite product measures, we provide sufficient conditions for the equivalence of laws of the two sequences. urn:nbn:de:0297-zib-60230 Numerical Mathematics
OPUS4-5942 misc Sullivan, T. J. Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priors 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. 2016 urn:nbn:de:0297-zib-59422 10.3934/ipi.2017040 Numerical Mathematics
OPUS4-5951 misc Cockayne, Jon; Oates, Chris; Sullivan, T. J.; Girolami, Mark Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems 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. urn:nbn:de:0297-zib-59513 Numerical Mathematics
OPUS4-6975 misc Nava-Yazdani, Esfandiar; Hege, Hans-Christian; von Tycowicz, Christoph; Sullivan, T. J. A Shape Trajectories Approach to Longitudinal Statistical Analysis For Kendall's shape space we determine analytically Jacobi fields and parallel transport, and compute geodesic regression. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and reduce the computational expense by several orders of magnitude. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As application example we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative. Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data only. urn:nbn:de:0297-zib-69759 Visual Data Analysis
OPUS4-6632 Wissenschaftlicher Artikel Lie, Han Cheng; Sullivan, T. J.; Teckentrup, Aretha Random forward models and log-likelihoods in Bayesian inverse problems 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. 29 SIAM/ASA Journal on Uncertainty Quantification 6 4 1600 1629 urn:nbn:de:0297-zib-66324 10.1137/18M1166523 Numerical Mathematics