@article{SchusterConstantineSullivan, author = {Schuster, Ingmar and Constantine, Paul and Sullivan, T. J.}, title = {Exact active subspace Metropolis-Hastings, with applications to the Lorenz-96 system}, abstract = {We consider the application of active subspaces to inform a Metropolis-Hastings algorithm, thereby aggressively reducing the computational dimension of the sampling problem. We show that the original formulation, as proposed by Constantine, Kent, and Bui-Thanh (SIAM J. Sci. Comput., 38(5):A2779-A2805, 2016), possesses asymptotic bias. Using pseudo-marginal arguments, we develop an asymptotically unbiased variant. Our algorithm is applied to a synthetic multimodal target distribution as well as a Bayesian formulation of a parameter inference problem for a Lorenz-96 system.}, 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} } @misc{Sullivan2016, author = {Sullivan, T. J.}, title = {Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priors}, issn = {1438-0064}, doi = {10.3934/ipi.2017040}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-59422}, year = {2016}, abstract = {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-Lo{\`e}ve 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.}, language = {en} } @misc{CockayneOatesSullivanetal., author = {Cockayne, Jon and Oates, Chris and Sullivan, T. J. and Girolami, Mark}, title = {Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-59513}, abstract = {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.}, 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{KlebanovSchusterSullivan, author = {Klebanov, Ilja and Schuster, Ingmar and Sullivan, T. J.}, title = {A rigorous theory of conditional mean embeddings}, series = {SIAM Journal on Mathematics of Data Science}, volume = {2}, journal = {SIAM Journal on Mathematics of Data Science}, number = {3}, doi = {10.1137/19M1305069}, pages = {583 -- 606}, language = {en} } @article{NavaYazdaniHegeSullivanetal., author = {Nava-Yazdani, Esfandiar and Hege, Hans-Christian and Sullivan, T. J. and von Tycowicz, Christoph}, title = {Geodesic Analysis in Kendall's Shape Space with Epidemiological Applications}, series = {Journal of Mathematical Imaging and Vision}, volume = {62}, journal = {Journal of Mathematical Imaging and Vision}, number = {4}, doi = {10.1007/s10851-020-00945-w}, pages = {549 -- 559}, abstract = {We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall's shape space. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude over common, nonlinear constrained approaches. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As an example application we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative (OAI). 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 alone.}, language = {en} } @article{KlebanovSprungkSullivan, author = {Klebanov, Ilja and Sprungk, Bj{\"o}rn and Sullivan, T. J.}, title = {The linear conditional expectation in Hilbert space}, series = {Bernoulli}, volume = {27}, journal = {Bernoulli}, number = {4}, doi = {10.3150/20-BEJ1308}, pages = {2299 -- 2299}, language = {en} } @article{BonnetAkianSavinetal., author = {Bonnet, Luc and Akian, Jean-Luc and Savin, {\´E}ric and Sullivan, T. J.}, title = {Adaptive reconstruction of imperfectly-observed monotone functions, with applications to uncertainty quantification}, series = {Algorithms}, volume = {13}, journal = {Algorithms}, number = {8}, doi = {10.3390/a13080196}, pages = {196}, abstract = {Motivated by the desire to numerically calculate rigorous upper and lower bounds on deviation probabilities over large classes of probability distributions, we present an adaptive algorithm for the reconstruction of increasing real-valued functions. While this problem is similar to the classical statistical problem of isotonic regression, the optimisation setting alters several characteristics of the problem and opens natural algorithmic possibilities. We present our algorithm, establish sufficient conditions for convergence of the reconstruction to the ground truth, and apply the method to synthetic test cases and a real-world example of uncertainty quantification for aerodynamic design.}, 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} }