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 - 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 - 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 - 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 - 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 - 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 - JOUR
A1 - Schuster, Ingmar
A1 - Constantine, Paul
A1 - Sullivan, T. J.
T1 - Exact active subspace Metropolisâ€“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 - 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 - 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 -