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
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 - 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 - 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
ER -