@article{AyanbayevKlebanovLieetal., author = {Ayanbayev, Birzhan and Klebanov, Ilja and Lie, Han Cheng and Sullivan, T. J.}, title = {Γ-convergence of Onsager-Machlup functionals: II. Infinite product measures on Banach spaces}, series = {Inverse Problems}, volume = {38}, journal = {Inverse Problems}, number = {2}, doi = {10.1088/1361-6420/ac3f82}, pages = {35}, language = {en} } @article{AyanbayevKlebanovLieetal., author = {Ayanbayev, Birzhan and Klebanov, Ilja and Lie, Han Cheng and Sullivan, T. J.}, title = {Γ-convergence of Onsager-Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems}, series = {Inverse Problems}, volume = {38}, journal = {Inverse Problems}, number = {2}, doi = {10.1088/1361-6420/ac3f81}, pages = {32}, language = {en} } @article{Sullivan, author = {Sullivan, T. J.}, title = {Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors}, series = {Inverse Problems and Imaging}, volume = {11}, journal = {Inverse Problems and Imaging}, number = {5}, doi = {10.3934/ipi.2017040}, pages = {857 -- 874}, 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{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{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} } @article{OwhadiScovelSullivan, author = {Owhadi, Houman and Scovel, Clint and Sullivan, T. J.}, title = {On the Brittleness of Bayesian Inference}, series = {SIAM Review}, volume = {57}, journal = {SIAM Review}, number = {4}, doi = {10.1137/130938633}, pages = {566 -- 582}, abstract = {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.}, 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{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{LieSullivanTeckentrup2020, author = {Lie, Han Cheng and Sullivan, T. J. and Teckentrup, Aretha}, title = {Error bounds for some approximate posterior measures in Bayesian inference}, series = {Numerical Mathematics and Advanced Applications ENUMATH 2019}, journal = {Numerical Mathematics and Advanced Applications ENUMATH 2019}, publisher = {Springer}, doi = {10.1007/978-3-030-55874-1_26}, pages = {275 -- 283}, year = {2020}, language = {en} }