TY - JOUR A1 - Ayanbayev, Birzhan A1 - Klebanov, Ilja A1 - Lie, Han Cheng A1 - Sullivan, T. J. T1 - Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces JF - Inverse Problems Y1 - 2022 U6 - https://doi.org/10.1088/1361-6420/ac3f82 VL - 38 IS - 2 ER - TY - JOUR A1 - Ayanbayev, Birzhan A1 - Klebanov, Ilja A1 - Lie, Han Cheng A1 - Sullivan, T. J. T1 - Γ-convergence of Onsager–Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems JF - Inverse Problems Y1 - 2022 U6 - https://doi.org/10.1088/1361-6420/ac3f81 VL - 38 IS - 2 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 - 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 - 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 - 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 - 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 - 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 - 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. 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 -