TY - JOUR
A1 - Oates, Chris. J.
A1 - Cockayne, Jon
A1 - Prangle, Dennis
A1 - Sullivan, T. J.
A1 - Girolami, Mark
T1 - Optimality criteria for probabilistic numerical methods
N2 - It is well understood that Bayesian decision theory and average case analysis are essentially identical. However, if one is interested in performing uncertainty quantification for a numerical task, it can be argued that the decision-theoretic framework is neither appropriate nor sufficient. To this end, we consider an alternative optimality criterion from Bayesian experimental design and study its implied optimal information in the numerical context. This information is demonstrated to differ, in general, from the information that would be used in an average-case-optimal numerical method. The explicit connection to Bayesian experimental design suggests several distinct regimes in which optimal probabilistic numerical methods can be developed.
Y1 - 2019
ER -
TY - JOUR
A1 - Oates, Chris. J.
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 - http://dx.doi.org/10.1007/s11222-019-09902-z
VL - 29
IS - 6
SP - 1335
EP - 1351
ER -
TY - JOUR
A1 - Girolami, Mark A.
A1 - Ipsen, Ilse C. F.
A1 - Oates, Chris J.
A1 - Owen, Art B.
A1 - Sullivan, T. J.
T1 - Editorial: Special edition on probabilistic numerics
JF - Statistics and Computing
Y1 - 2019
U6 - http://dx.doi.org/doi:10.1007/s11222-019-09892-y
VL - 29
IS - 6
SP - 1181
EP - 1183
ER -
TY - JOUR
A1 - Lie, Han Cheng
A1 - Sullivan, T. J.
T1 - Cameron--Martin theorems for sequences of Cauchy-distributed random variables
JF - arXiv
N2 - 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.
Y1 - 2016
SP - 1608.03784
ER -
TY - GEN
A1 - Lie, Han Cheng
A1 - Sullivan, T. J.
T1 - Cameron--Martin theorems for sequences of Cauchy-distributed random variables
N2 - 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.
T3 - ZIB-Report - 16-40
Y1 - 2016
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-60230
SN - 1438-0064
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 - 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 - http://dx.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 - http://dx.doi.org/10.1137/130938633
VL - 57
IS - 4
SP - 566
EP - 582
ER -
TY - BOOK
A1 - Sullivan, T. J.
T1 - Introduction to Uncertainty Quantification
Y1 - 2015
SN - 978-3-319-23394-9
U6 - http://dx.doi.org/10.1007/978-3-319-23395-6
VL - 63
PB - Springer
ER -
TY - CHAP
A1 - Cockayne, Jon
A1 - Oates, Chris
A1 - Sullivan, T. J.
A1 - Girolami, Mark
T1 - Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems
T2 - Proceedings of the 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering
N2 - This paper develops meshless methods for probabilistically describing discretisation error in the numerical solution of partial differential equations. This construction enables the solution of Bayesian inverse problems while accounting for the impact of the discretisation of the forward problem. In particular, this drives statistical inferences to be more conservative in the presence of significant solver error. Theoretical results are presented describing rates of convergence for the posteriors in both the forward and inverse problems. This method is tested on a challenging inverse problem with a nonlinear forward model
Y1 - 2017
U6 - http://dx.doi.org/10.1063/1.4985359
VL - 1853
SP - 060001-1
EP - 060001-8
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
Y1 - 2017
ER -
TY - JOUR
A1 - Lie, Han Cheng
A1 - Sullivan, T. J.
T1 - Equivalence of weak and strong modes of measures on topological vector spaces
JF - Inverse Problems
N2 - 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.
Y1 - 2018
U6 - http://dx.doi.org/10.1088/1361-6420/aadef2
VL - 34
IS - 11
SP - 115013
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.
T1 - Erratum: Equivalence of weak and strong modes of measures on topological vector spaces (2018 Inverse Problems 34 115013)
JF - Inverse Problems
Y1 - 2018
U6 - http://dx.doi.org/10.1088/1361-6420/aae55b
VL - 34
IS - 12
SP - 129601
ER -
TY - JOUR
A1 - Kersting, Hans
A1 - Sullivan, T. J.
A1 - Hennig, Philipp
T1 - Convergence rates of Gaussian ODE filters
Y1 - 2018
ER -
TY - CHAP
A1 - Sullivan, T. J.
T1 - Well-posedness of Bayesian inverse problems in quasi-Banach spaces with stable priors
T2 - 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Weimar 2017
Y1 - 2017
U6 - http://dx.doi.org/10.1002/pamm.201710402
VL - 17
IS - 1
SP - 871
EP - 874
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 - http://dx.doi.org/10.3934/ipi.2017040
VL - 11
IS - 5
SP - 857
EP - 874
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 - http://dx.doi.org/10.1214/19-BA1145
VL - 14
IS - 3
SP - 985
EP - 989
ER -
TY - GEN
A1 - Nava-Yazdani, Esfandiar
A1 - Hege, Hans-Christian
A1 - von Tycowicz, Christoph
A1 - Sullivan, T. J.
T1 - A Shape Trajectories Approach to Longitudinal Statistical Analysis
N2 - For Kendall’s shape space we determine analytically Jacobi fields and parallel transport, and compute geodesic regression. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and reduce the computational expense by several orders of magnitude. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data.
As application example we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative. 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 only.
T3 - ZIB-Report - 18-42
KW - shape space, shape trajectories, geodesic regression, longitudinal analysis, osteoarthritis
Y1 - 2018
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-69759
SN - 1438-0064
ER -
TY - JOUR
A1 - Lie, Han Cheng
A1 - Sullivan, T. J.
A1 - Teckentrup, Aretha
T1 - Random forward models and log-likelihoods in Bayesian inverse problems
JF - SIAM/ASA Journal on Uncertainty Quantification
N2 - We consider the use of randomised forward models and log-likelihoods within the Bayesian approach to inverse problems. Such random approximations to the exact forward model or log-likelihood arise naturally when a computationally expensive model is approximated using a cheaper stochastic surrogate, as in Gaussian process emulation (kriging), or in the field of probabilistic numerical methods. We show that the Hellinger distance between the exact and approximate Bayesian posteriors is bounded by moments of the difference between the true and approximate log-likelihoods. Example applications of these stability results are given for randomised misfit models in large data applications and the probabilistic solution of ordinary differential equations.
T3 - ZIB-Report - 18-03
KW - Bayesian inverse problem
KW - random likelihood
KW - surrogate model
KW - posterior consistency
KW - probabilistic numerics
KW - uncertainty quantification
KW - randomised misfit
Y1 - 2018
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-66324
SN - 1438-0064
VL - 6
IS - 4
SP - 1600
EP - 1629
ER -
TY - CHAP
A1 - Teymur, Onur
A1 - Lie, Han Cheng
A1 - Sullivan, T. J.
A1 - Calderhead, Ben
T1 - Implicit probabilistic integrators for ODEs
T2 - Advances in Neural Information Processing Systems 31 (NIPS 2018)
Y1 - 2018
UR - http://papers.nips.cc/paper/7955-implicit-probabilistic-integrators-for-odes
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 - http://dx.doi.org/10.1007/s11222-019-09898-6
VL - 29
IS - 6
SP - 1265
EP - 1283
ER -
TY - JOUR
A1 - Cockayne, Jon
A1 - Oates, Chris
A1 - Sullivan, T. J.
A1 - Girolami, Mark
T1 - Bayesian Probabilistic Numerical Methods
JF - SIAM Review
N2 - 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.
Y1 - 2019
U6 - http://dx.doi.org/10.1137/17M1139357
VL - 61
IS - 4
SP - 756
EP - 789
ER -
TY - JOUR
A1 - Klebanov, Ilja
A1 - Schuster, Ingmar
A1 - Sullivan, T. J.
T1 - A rigorous theory of conditional mean embeddings
Y1 - 2019
ER -