Differentiable Likelihoods for Fast Inversion of ’Likelihood-Free’ Dynamical Systems
Hans Kersting, Nicholas Krämer, Martin Schiegg, Christian Daniel, Michael Tiemann, Philipp Hennig
Abstract
Likelihood-free (a.k.a. simulation-based) inference problems are inverse problems with expensive, or intractable, forward models. ODE inverse problems are commonly treated as likelihood-free, as their forward map has to be numerically approximated by an ODE solver. This, however, is not a fundamental constraint but just a lack of functionality in classic ODE solvers, which do not return a likelihood but a point estimate. To address this shortcoming, we employ Gaussian ODE filtering (a probabilistic numerical method for ODEs) to construct a local Gaussian approximation to the likelihood. This approximation yields tractable estimators for the gradient and Hessian of the (log-)likelihood. Insertion of these estimators into existing gradient-based optimization and sampling methods engenders new solvers for ODE inverse problems. We demonstrate that these methods outperform standard likelihood-free approaches on three benchmark-systems.
BibTeX
@InProceedings{pmlr-v119-kersting20a,
title = {Differentiable Likelihoods for Fast Inversion of ’{L}ikelihood-Free’ Dynamical Systems},
author = {Kersting, Hans and Kr{\"a}mer, Nicholas and Schiegg, Martin and Daniel, Christian and Tiemann, Michael and Hennig, Philipp},
booktitle = {Proceedings of the 37th International Conference on Machine Learning},
pages = {5198--5208},
year = {2020},
editor = {III, Hal Daumé and Singh, Aarti},
volume = {119},
series = {Proceedings of Machine Learning Research},
month = {13--18 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v119/kersting20a/kersting20a.pdf},
url = {https://proceedings.mlr.press/v119/kersting20a.html},
abstract = {Likelihood-free (a.k.a. simulation-based) inference problems are inverse problems with expensive, or intractable, forward models. ODE inverse problems are commonly treated as likelihood-free, as their forward map has to be numerically approximated by an ODE solver. This, however, is not a fundamental constraint but just a lack of functionality in classic ODE solvers, which do not return a likelihood but a point estimate. To address this shortcoming, we employ Gaussian ODE filtering (a probabilistic numerical method for ODEs) to construct a local Gaussian approximation to the likelihood. This approximation yields tractable estimators for the gradient and Hessian of the (log-)likelihood. Insertion of these estimators into existing gradient-based optimization and sampling methods engenders new solvers for ODE inverse problems. We demonstrate that these methods outperform standard likelihood-free approaches on three benchmark-systems.}
}