@misc{LieSullivanTeckentrup, author = {Lie, Han Cheng and Sullivan, T. J. and Teckentrup, Aretha}, title = {Random forward models and log-likelihoods in Bayesian inverse problems}, series = {SIAM/ASA Journal on Uncertainty Quantification}, volume = {6}, journal = {SIAM/ASA Journal on Uncertainty Quantification}, number = {4}, issn = {1438-0064}, doi = {10.1137/18M1166523}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-66324}, pages = {1600 -- 1629}, abstract = {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.}, language = {en} } @article{LieSullivan, author = {Lie, Han Cheng and Sullivan, T. J.}, title = {Equivalence of weak and strong modes of measures on topological vector spaces}, series = {Inverse Problems}, volume = {34}, journal = {Inverse Problems}, number = {11}, doi = {10.1088/1361-6420/aadef2}, pages = {115013}, abstract = {Modes of a probability measure on an infinite-dimensional Banach space X are often defined by maximising the small-radius limit of the ratio of measures of norm balls. Helin and Burger weakened the definition of such modes by considering only balls with centres in proper subspaces of X, and posed the question of when this restricted notion coincides with the unrestricted one. We generalise these definitions to modes of arbitrary measures on topological vector spaces, defined by arbitrary bounded, convex, neighbourhoods of the origin. We show that a coincident limiting ratios condition is a necessary and sufficient condition for the equivalence of these two types of modes, and show that the coincident limiting ratios condition is satisfied in a wide range of topological vector spaces.}, language = {en} } @article{LieSullivan2018, author = {Lie, Han Cheng and Sullivan, T. J.}, title = {Quasi-invariance of countable products of Cauchy measures under non-unitary dilations}, series = {Electronic Communications in Probability}, volume = {23}, journal = {Electronic Communications in Probability}, number = {8}, doi = {10.1214/18-ECP113}, pages = {1 -- 6}, year = {2018}, language = {en} } @inproceedings{TeymurLieSullivanetal., author = {Teymur, Onur and Lie, Han Cheng and Sullivan, T. J. and Calderhead, Ben}, title = {Implicit probabilistic integrators for ODEs}, series = {Advances in Neural Information Processing Systems 31 (NIPS 2018)}, booktitle = {Advances in Neural Information Processing Systems 31 (NIPS 2018)}, language = {en} } @article{LieSullivan, author = {Lie, Han Cheng and Sullivan, T. J.}, title = {Erratum: Equivalence of weak and strong modes of measures on topological vector spaces (2018 Inverse Problems 34 115013)}, series = {Inverse Problems}, volume = {34}, journal = {Inverse Problems}, number = {12}, doi = {10.1088/1361-6420/aae55b}, pages = {129601}, language = {en} } @misc{NavaYazdaniHegevonTycowiczetal., author = {Nava-Yazdani, Esfandiar and Hege, Hans-Christian and von Tycowicz, Christoph and Sullivan, T. J.}, title = {A Shape Trajectories Approach to Longitudinal Statistical Analysis}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-69759}, abstract = {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.}, language = {en} } @article{CockayneOatesSullivanetal., author = {Cockayne, Jon and Oates, Chris and Sullivan, T. J. and Girolami, Mark}, title = {Bayesian Probabilistic Numerical Methods}, series = {SIAM Review}, volume = {61}, journal = {SIAM Review}, number = {4}, doi = {10.1137/17M1139357}, pages = {756 -- 789}, abstract = {Over forty years ago average-case error was proposed in the applied mathematics literature as an alternative criterion with which to assess numerical methods. In contrast to worst-case error, this criterion relies on the construction of a probability measure over candidate numerical tasks, and numerical methods are assessed based on their average performance over those tasks with respect to the measure. This paper goes further and establishes Bayesian probabilistic numerical methods as solutions to certain inverse problems based upon the numerical task within the Bayesian framework. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well defined, encompassing both the non-linear and non-Gaussian context. For general computation, a numerical approximation scheme is proposed and its asymptotic convergence established. The theoretical development is extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, and a challenging industrial application is presented.}, language = {en} } @article{LieSullivanStuart, author = {Lie, Han Cheng and Sullivan, T. J. and Stuart, Andrew}, title = {Strong convergence rates of probabilistic integrators for ordinary differential equations}, series = {Statistics and Computing}, volume = {29}, journal = {Statistics and Computing}, number = {6}, doi = {10.1007/s11222-019-09898-6}, pages = {1265 -- 1283}, abstract = {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.}, language = {en} } @article{GirolamiIpsenOatesetal., author = {Girolami, Mark A. and Ipsen, Ilse C. F. and Oates, Chris and Owen, Art B. and Sullivan, T. J.}, title = {Editorial: Special edition on probabilistic numerics}, series = {Statistics and Computing}, volume = {29}, journal = {Statistics and Computing}, number = {6}, doi = {doi:10.1007/s11222-019-09892-y}, pages = {1181 -- 1183}, language = {en} } @article{Sullivan, author = {Sullivan, T. J.}, title = {Contributed discussion on the article "A Bayesian conjugate gradient method"}, series = {Bayesian Analysis}, volume = {14}, journal = {Bayesian Analysis}, number = {3}, doi = {10.1214/19-BA1145}, pages = {985 -- 989}, abstract = {The recent article "A Bayesian conjugate gradient method" by Cockayne, Oates, Ipsen, and Girolami proposes an approximately Bayesian iterative procedure for the solution of a system of linear equations, based on the conjugate gradient method, that gives a sequence of Gaussian/normal estimates for the exact solution. The purpose of the probabilistic enrichment is that the covariance structure is intended to provide a posterior measure of uncertainty or confidence in the solution mean. This note gives some comments on the article, poses some questions, and suggests directions for further research.}, language = {en} }