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Active Uncertainty Calibration in Bayesian ODE Solvers

2016

Conference Paper

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There is resurging interest, in statistics and machine learning, in solvers for ordinary differential equations (ODEs) that return probability measures instead of point estimates. Recently, Conrad et al.~introduced a sampling-based class of methods that are `well-calibrated' in a specific sense. But the computational cost of these methods is significantly above that of classic methods. On the other hand, Schober et al.~pointed out a precise connection between classic Runge-Kutta ODE solvers and Gaussian filters, which gives only a rough probabilistic calibration, but at negligible cost overhead. By formulating the solution of ODEs as approximate inference in linear Gaussian SDEs, we investigate a range of probabilistic ODE solvers, that bridge the trade-off between computational cost and probabilistic calibration, and identify the inaccurate gradient measurement as the crucial source of uncertainty. We propose the novel filtering-based method Bayesian Quadrature filtering (BQF) which uses Bayesian quadrature to actively learn the imprecision in the gradient measurement by collecting multiple gradient evaluations.

Author(s): Kersting, H. and Hennig, P.
Book Title: Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence (UAI 2016)
Pages: 309--318
Year: 2016
Month: June
Editors: Ihler, A. and Janzing, D.
Publisher: {AUAI} Press

Department(s): Empirical Inference, Probabilistic Numerics
Research Project(s): Probabilistic Solvers for Ordinary Differential Equations
Bibtex Type: Conference Paper (conference)

Event Place: New York, USA

State: Published
URL: http://www.auai.org/uai2016/proceedings/papers/163.pdf

BibTex

@conference{KerHen16,
  title = {Active Uncertainty Calibration in Bayesian ODE Solvers},
  author = {Kersting, H. and Hennig, P.},
  booktitle = {Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence (UAI 2016)},
  pages = {309--318},
  editors = {Ihler, A. and Janzing, D.},
  publisher = {{AUAI} Press},
  month = jun,
  year = {2016},
  url = {http://www.auai.org/uai2016/proceedings/papers/163.pdf},
  month_numeric = {6}
}