  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.
