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Active Uncertainty Calibration in Bayesian ODE Solvers


Conference Paper



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)
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


  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)},
  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}