ODE filters and smoothers are well-established probabilistic numerical methods that solve initial value problems in linear time. In this talk, we add to Monday’s talks on these methods in the following way. First, we discuss how a previous state space model can be thought of as a linear-Gaussian approximation to the new state space model. Second, we discuss classical convergence rates for the integrated-Wiener process prior-—-as well as equivalences with classical methods and their convergence rates. Third, we show how ODE filters give speed-ups in ODE inverse problems, a first instance of a computational chain communicating uncertainty.