@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{LieSullivan, author = {Lie, Han Cheng and Sullivan, T. J.}, title = {Cameron--Martin theorems for sequences of Cauchy-distributed random variables}, series = {arXiv}, journal = {arXiv}, pages = {1608.03784}, abstract = {Given a sequence of Cauchy-distributed random variables defined by a sequence of location parameters and a sequence of scale parameters, we consider another sequence of random variables that is obtained by perturbing the location or scale parameter sequences. Using a result of Kakutani on equivalence of infinite product measures, we provide sufficient conditions for the equivalence of laws of the two sequences.}, language = {en} } @article{CockayneOatesSullivanetal., author = {Cockayne, Jon and Oates, Chris and Sullivan, T. J. and Girolami, Mark}, title = {Bayesian Probabilistic Numerical Methods}, series = {SIAM Review}, volume = {61}, journal = {SIAM Review}, number = {4}, doi = {10.1137/17M1139357}, pages = {756 -- 789}, abstract = {Over forty years ago average-case error was proposed in the applied mathematics literature as an alternative criterion with which to assess numerical methods. In contrast to worst-case error, this criterion relies on the construction of a probability measure over candidate numerical tasks, and numerical methods are assessed based on their average performance over those tasks with respect to the measure. This paper goes further and establishes Bayesian probabilistic numerical methods as solutions to certain inverse problems based upon the numerical task within the Bayesian framework. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well defined, encompassing both the non-linear and non-Gaussian context. For general computation, a numerical approximation scheme is proposed and its asymptotic convergence established. The theoretical development is extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, and a challenging industrial application is presented.}, 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{SchaeferSullivanOwhadi, author = {Sch{\"a}fer, Florian and Sullivan, T. J. and Owhadi, Houman}, title = {Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity}, series = {Multiscale Modeling and Simulation}, volume = {19}, journal = {Multiscale Modeling and Simulation}, number = {2}, doi = {10.1137/19M129526X}, pages = {688 -- 730}, language = {en} } @article{LieSullivan, author = {Lie, Han Cheng and Sullivan, T. J.}, title = {Equivalence of weak and strong modes of measures on topological vector spaces}, series = {Inverse Problems}, volume = {34}, journal = {Inverse Problems}, number = {11}, doi = {10.1088/1361-6420/aadef2}, pages = {115013}, abstract = {Modes of a probability measure on an infinite-dimensional Banach space X are often defined by maximising the small-radius limit of the ratio of measures of norm balls. Helin and Burger weakened the definition of such modes by considering only balls with centres in proper subspaces of X, and posed the question of when this restricted notion coincides with the unrestricted one. We generalise these definitions to modes of arbitrary measures on topological vector spaces, defined by arbitrary bounded, convex, neighbourhoods of the origin. We show that a coincident limiting ratios condition is a necessary and sufficient condition for the equivalence of these two types of modes, and show that the coincident limiting ratios condition is satisfied in a wide range of topological vector spaces.}, 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{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{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} }