  We formulate probabilistic numerical approximations to solutions of ordinary
differential equations (ODEs) as problems in Gaussian process (GP) regression
with non-linear measurement functions. This is achieved by defining the
measurement sequence to consist of the observations of the difference between
the derivative of the GP and the vector field evaluated at the GP---which are
all identically zero at the solution of the ODE. When the GP has a state-space
representation, the problem can be reduced to a non-linear Bayesian filtering
problem and all widely-used approximations to the Bayesian filtering and
smoothing problems become applicable. Furthermore, all previous GP-based ODE
solvers that are formulated in terms of generating synthetic measurements of
the gradient field come out as specific approximations. Based on the non-linear
Bayesian filtering problem posed in this paper, we develop novel Gaussian
solvers for which we establish favourable stability properties. Additionally,
non-Gaussian approximations to the filtering problem are derived by the
particle filter approach. The resulting solvers are compared with other
probabilistic solvers in illustrative experiments.
