Differentiable Likelihoods for Fast Inversion of 'Likelihood-Free' Dynamical Systems

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.

11 pages (+ 5 pages appendix), 6 figures

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Cited by in corpus (4)

• Variational Autoencoding of PDE Inverse Problems
• A Probabilistic State Space Model for Joint Inference from Differential Equations and Data
• A Fourier State Space Model for Bayesian ODE Filters
• Linear-Time Probabilistic Solutions of Boundary Value Problems