@article{LieSullivan,
author = {Lie, Han Cheng and Sullivan, T. J.},
title = {Cameron--Martin theorems for sequences of Cauchy-distributed random variables},
series = {arXiv},
journal = {arXiv},
pages = {1608.03784},
abstract = {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.},
language = {en}
}
@misc{LieSullivan,
author = {Lie, Han Cheng and Sullivan, T. J.},
title = {Cameron--Martin theorems for sequences of Cauchy-distributed random variables},
issn = {1438-0064},
url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-60230},
abstract = {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.},
language = {en}
}
@article{LieSullivan2018,
author = {Lie, Han Cheng and Sullivan, T. J.},
title = {Quasi-invariance of countable products of Cauchy measures under non-unitary dilations},
series = {Electronic Communications in Probability},
volume = {23},
journal = {Electronic Communications in Probability},
number = {8},
doi = {10.1214/18-ECP113},
pages = {1 -- 6},
year = {2018},
language = {en}
}
@article{LieSullivan,
author = {Lie, Han Cheng and Sullivan, T. J.},
title = {Equivalence of weak and strong modes of measures on topological vector spaces},
series = {Inverse Problems},
volume = {34},
journal = {Inverse Problems},
number = {11},
doi = {10.1088/1361-6420/aadef2},
pages = {115013},
abstract = {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.},
language = {en}
}
@article{LieSullivan,
author = {Lie, Han Cheng and Sullivan, T. J.},
title = {Erratum: Equivalence of weak and strong modes of measures on topological vector spaces (2018 Inverse Problems 34 115013)},
series = {Inverse Problems},
volume = {34},
journal = {Inverse Problems},
number = {12},
doi = {10.1088/1361-6420/aae55b},
pages = {129601},
language = {en}
}
@article{LieSullivanTeckentrup,
author = {Lie, Han Cheng and Sullivan, T. J. and Teckentrup, Aretha},
title = {Random forward models and log-likelihoods in Bayesian inverse problems},
series = {SIAM/ASA Journal on Uncertainty Quantification},
volume = {6},
journal = {SIAM/ASA Journal on Uncertainty Quantification},
number = {4},
issn = {1438-0064},
doi = {10.1137/18M1166523},
url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-66324},
pages = {1600 -- 1629},
abstract = {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.},
language = {en}
}
@inproceedings{TeymurLieSullivanetal.,
author = {Teymur, Onur and Lie, Han Cheng and Sullivan, T. J. and Calderhead, Ben},
title = {Implicit probabilistic integrators for ODEs},
series = {Advances in Neural Information Processing Systems 31 (NIPS 2018)},
booktitle = {Advances in Neural Information Processing Systems 31 (NIPS 2018)},
language = {en}
}
@article{LieSullivanStuart,
author = {Lie, Han Cheng and Sullivan, T. J. and Stuart, Andrew},
title = {Strong convergence rates of probabilistic integrators for ordinary differential equations},
series = {Statistics and Computing},
volume = {29},
journal = {Statistics and Computing},
number = {6},
doi = {10.1007/s11222-019-09898-6},
pages = {1265 -- 1283},
abstract = {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.},
language = {en}
}