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 - 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 -