TY - JOUR A1 - Wang, Junyang A1 - Cockayne, Jon A1 - Chkrebtii, Oksana A1 - Sullivan, T. J. A1 - Oates, Chris T1 - Bayesian numerical methods for nonlinear partial differential equations JF - Statistics and Computing Y1 - 2021 U6 - https://doi.org/10.1007/s11222-021-10030-w VL - 31 IS - 5 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 - 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 - JOUR A1 - Schäfer, Florian A1 - Sullivan, T. J. A1 - Owhadi, Houman T1 - Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity JF - Multiscale Modeling and Simulation Y1 - 2021 U6 - https://doi.org/10.1137/19M129526X VL - 19 IS - 2 SP - 688 EP - 730 ER - 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 - Owhadi, Houman A1 - Scovel, Clint A1 - Sullivan, T. J. T1 - On the Brittleness of Bayesian Inference JF - SIAM Review N2 - With the advent of high-performance computing, Bayesian methods are becoming increasingly popular tools for the quantification of uncertainty throughout science and industry. Since these methods can impact the making of sometimes critical decisions in increasingly complicated contexts, the sensitivity of their posterior conclusions with respect to the underlying models and prior beliefs is a pressing question to which there currently exist positive and negative answers. We report new results suggesting that, although Bayesian methods are robust when the number of possible outcomes is finite or when only a finite number of marginals of the data-generating distribution are unknown, they could be generically brittle when applied to continuous systems (and their discretizations) with finite information on the data-generating distribution. If closeness is defined in terms of the total variation (TV) metric or the matching of a finite system of generalized moments, then (1) two practitioners who use arbitrarily close models and observe the same (possibly arbitrarily large amount of) data may reach opposite conclusions; and (2) any given prior and model can be slightly perturbed to achieve any desired posterior conclusion. The mechanism causing brittleness/robustness suggests that learning and robustness are antagonistic requirements, which raises the possibility of a missing stability condition when using Bayesian inference in a continuous world under finite information. Y1 - 2015 U6 - https://doi.org/10.1137/130938633 VL - 57 IS - 4 SP - 566 EP - 582 ER - TY - JOUR A1 - Oates, Chris A1 - Sullivan, T. J. T1 - A modern retrospective on probabilistic numerics JF - Statistics and Computing N2 - This article attempts to place the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related to modern formal treatments and applications. We highlight in particular the parallel contributions of Sul'din and Larkin in the 1960s and how their pioneering early ideas have reached a degree of maturity in the intervening period, mediated by paradigms such as average-case analysis and information-based complexity. We provide a subjective assessment of the state of research in probabilistic numerics and highlight some difficulties to be addressed by future works. Y1 - 2019 U6 - https://doi.org/10.1007/s11222-019-09902-z VL - 29 IS - 6 SP - 1335 EP - 1351 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 - Lie, Han Cheng A1 - Sullivan, T. J. A1 - Teckentrup, Aretha T1 - Error bounds for some approximate posterior measures in Bayesian inference JF - Numerical Mathematics and Advanced Applications ENUMATH 2019 Y1 - 2020 U6 - https://doi.org/10.1007/978-3-030-55874-1_26 SP - 275 EP - 283 PB - Springer 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 -