I am Junior Professor in Applied Mathematics with Specialism in Risk and Uncertainty Quantification at the Free University of Berlin and Research Group Leader for Uncertainty Quantification at the Zuse Institute Berlin. I have wide interests in uncertainty quantification the broad sense, understood as the meeting point of numerical analysis, applied probability and statistics, and scientific computation. On this site you will find information about how to contact me, my research, publications, and teaching activities.
Chris Oates and I have just uploaded a preprint of our paper “A modern retrospective on probabilistic numerics” to the arXiv.
Abstract. This article attempts to cast 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.
Chris Oates, Jon Cockayne, Dennis Prangle, Mark Girolami, and I have just uploaded a preprint of our paper “Optimality criteria for probabilistic numerical methods” to the arXiv.
Abstract. 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.
SIAM/ASA Journal on Uncertainty Quantification (JUQ) publishes research articles presenting significant mathematical, statistical, algorithmic, and application advances in uncertainty quantification, defined as the interface of complex modeling of processes and data, especially characterizations of the uncertainties inherent in the use of such models. The journal also focuses on related fields such as sensitivity analysis, model validation, model calibration, data assimilation, and code verification. The journal also solicits papers describing new ideas that could lead to significant progress in methodology for uncertainty quantification as well as review articles on particular aspects. The journal is dedicated to nurturing synergistic interactions between the mathematical, statistical, computational, and applications communities involved in uncertainty quantification and related areas. JUQ is jointly offered by SIAM and the American Statistical Association.
The paper “Implicit probabilistic integrators for ODEs” by Onur Teymur, Han Cheng Lie, Ben Calderhead and myself has now appeared in Advances in Neural Information Processing Systems 31 (NIPS 2018). This paper forms part of an expanding body of work that provides mathematical convergence analysis of probabilistic solvers for initial value problems, in this case implicit methods such as (probabilistic versions of) the multistep Adams–Moulton method.
O. Teymur, H. C. Lie, T. J. Sullivan & B. Calderhead. “Implicit probabilistic integrators for ODEs” in Advances in Neural Information Processing Systems 31 (NIPS 2018), ed. N. Cesa-Bianchi, K. Grauman, H. Larochelle & H. Wallach. 2018. http://papers.nips.cc/paper/7955-implicit-probabilistic-integrators-for-odes
Abstract. We introduce a family of implicit probabilistic integrators for initial value problems (IVPs), taking as a starting point the multistep Adams–Moulton method. The implicit construction allows for dynamic feedback from the forthcoming time-step, in contrast to previous probabilistic integrators, all of which are based on explicit methods. We begin with a concise survey of the rapidly-expanding field of probabilistic ODE solvers. We then introduce our method, which builds on and adapts the work of Conrad et al. (2016) and Teymur et al. (2016), and provide a rigorous proof of its well-definedness and convergence. We discuss the problem of the calibration of such integrators and suggest one approach. We give an illustrative example highlighting the effect of the use of probabilistic integrators — including our new method — in the setting of parameter inference within an inverse problem.
The article “Random forward models and log-likelihoods in Bayesian inverse problems” by Han Cheng Lie, Aretha Teckentrup, and myself has now appeared in its final form in the SIAM/ASA Journal on Uncertainty Quantification, volume 6, issue 4. This paper considers the effect of approximating the likelihood in a Bayesian inverse problem by a random surrogate, as frequently happens in applications, with the aim of showing that the perturbed posterior distribution is close to the exact one in a suitable sense. This article considers general randomisation models, and thereby expands upon the previous investigations of Stuart and Teckentrup (2017) in the Gaussian setting.
H. C. Lie, T. J. Sullivan & A. L. Teckentrup. “Random forward models and log-likelihoods in Bayesian inverse problems.” SIAM/ASA Journal on Uncertainty Quantification 6(4):1600–1629, 2018. doi:10.1137/18M1166523
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.