TY - JOUR A1 - Schuster, Ingmar A1 - Constantine, Paul A1 - Sullivan, T. J. T1 - Exact active subspace Metropolis–Hastings, with applications to the Lorenz-96 system N2 - 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. Y1 - 2017 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 - 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 - 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 - JOUR A1 - Girolami, Mark A. A1 - Ipsen, Ilse C. F. A1 - Oates, Chris A1 - Owen, Art B. A1 - Sullivan, T. J. T1 - Editorial: Special edition on probabilistic numerics JF - Statistics and Computing Y1 - 2019 U6 - https://doi.org/doi:10.1007/s11222-019-09892-y VL - 29 IS - 6 SP - 1181 EP - 1183 ER - TY - JOUR A1 - Klebanov, Ilja A1 - Schuster, Ingmar A1 - Sullivan, T. J. T1 - A rigorous theory of conditional mean embeddings JF - SIAM Journal on Mathematics of Data Science Y1 - 2020 U6 - https://doi.org/10.1137/19M1305069 VL - 2 IS - 3 SP - 583 EP - 606 ER - TY - JOUR A1 - Nava-Yazdani, Esfandiar A1 - Hege, Hans-Christian A1 - Sullivan, T. J. A1 - von Tycowicz, Christoph T1 - Geodesic Analysis in Kendall's Shape Space with Epidemiological Applications JF - Journal of Mathematical Imaging and Vision N2 - 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. Y1 - 2020 U6 - https://doi.org/10.1007/s10851-020-00945-w VL - 62 IS - 4 SP - 549 EP - 559 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 - Bonnet, Luc A1 - Akian, Jean-Luc A1 - Savin, Éric A1 - Sullivan, T. J. T1 - Adaptive reconstruction of imperfectly-observed monotone functions, with applications to uncertainty quantification JF - Algorithms N2 - 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. Y1 - 2020 U6 - https://doi.org/10.3390/a13080196 VL - 13 IS - 8 SP - 196 ER - TY - JOUR A1 - Sullivan, T. J. T1 - Contributed discussion on the article "A Bayesian conjugate gradient method" JF - Bayesian Analysis N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1214/19-BA1145 VL - 14 IS - 3 SP - 985 EP - 989 ER -