6014
2016
eng
1608.03784
article
0
--
--
--
Cameron--Martin theorems for sequences of Cauchy-distributed random variables
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.
arXiv
1608.03784
no
Han Cheng Lie
Han Cheng Lie
T. J. Sullivan
Numerical Mathematics
Sullivan, Tim
no-project
Lie, Han
6023
eng
reportzib
0
--
2016-08-23
--
Cameron--Martin theorems for sequences of Cauchy-distributed random variables
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.
1438-0064
urn:nbn:de:0297-zib-60230
Han Cheng Lie
Han Cheng Lie
T. J. Sullivan
ZIB-Report
16-40
PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX)
Numerical Mathematics
Sullivan, Tim
no-project
Lie, Han
https://opus4.kobv.de/opus4-zib/files/6023/ZIB_report.pdf
6241
2018
2018
eng
1
6
6
8
23
article
0
2018-02-21
--
--
Quasi-invariance of countable products of Cauchy measures under non-unitary dilations
Electronic Communications in Probability
1611.10289
10.1214/18-ECP113
Yes
Han Cheng Lie
Regine Kossick
T. J. Sullivan
Numerical Mathematics
Sullivan, Tim
no-project
6660
2018
eng
115013
11
34
article
0
--
--
--
Equivalence of weak and strong modes of measures on topological vector spaces
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.
Inverse Problems
1708.02516
10.1088/1361-6420/aadef2
yes
Han Cheng Lie
Tim Sullivan
T. J. Sullivan
Numerical Mathematics
Sullivan, Tim
no-project
Lie, Han
7105
2018
eng
129601
12
34
article
0
--
--
--
Erratum: Equivalence of weak and strong modes of measures on topological vector spaces (2018 Inverse Problems 34 115013)
Inverse Problems
10.1088/1361-6420/aae55b
yes
2018
Han Cheng Lie
Tim Sullivan
T. J. Sullivan
Numerical Mathematics
Sullivan, Tim
no-project
Lie, Han
6632
eng
1600
1629
4
6
article
0
--
2018-01-04
--
Random forward models and log-likelihoods in Bayesian inverse problems
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.
SIAM/ASA Journal on Uncertainty Quantification
1438-0064
urn:nbn:de:0297-zib-66324
1712.05717
10.1137/18M1166523
2018
yes
Han Cheng Lie
Tim Sullivan
T. J. Sullivan
Aretha Teckentrup
ZIB-Report
18-03
eng
uncontrolled
Bayesian inverse problem
eng
uncontrolled
random likelihood
eng
uncontrolled
surrogate model
eng
uncontrolled
posterior consistency
eng
uncontrolled
probabilistic numerics
eng
uncontrolled
uncertainty quantification
eng
uncontrolled
randomised misfit
STATISTICS
NUMERICAL ANALYSIS
COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section -04 in that area)
Numerical Mathematics
Sullivan, Tim
SFB1114 A06
https://opus4.kobv.de/opus4-zib/files/6632/random-bip.pdf
6934
2018
eng
conferenceobject
0
--
--
--
Implicit probabilistic integrators for ODEs
Advances in Neural Information Processing Systems 31 (NIPS 2018)
1805.07970
http://papers.nips.cc/paper/7955-implicit-probabilistic-integrators-for-odes
2018
yes
Onur Teymur
Tim Sullivan
Han Cheng Lie
T. J. Sullivan
Ben Calderhead
Numerical Mathematics
Sullivan, Tim
no-project
6657
2019
eng
1265
1283
6
29
article
0
--
--
--
Strong convergence rates of probabilistic integrators for ordinary differential equations
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.
Statistics and Computing
1703.03680
10.1007/s11222-019-09898-6
yes
Han Cheng Lie
Tim Sullivan
T. J. Sullivan
Andrew Stuart
Numerical Mathematics
Sullivan, Tim
no-project
Lie, Han