Like many numerical methods, solvers for initial value problems (IVPs) on
ordinary differential equations estimate an analytically intractable quantity,
using the results of tractable computations as inputs. This structure is
closely connected to the notion of inference on latent variables in statistics...
We describe a class of algorithms that formulate the solution to an IVP as
inference on a latent path that is a draw from a Gaussian process probability
measure (or equivalently, the solution of a linear stochastic differential
equation). We then show that certain members of this class are connected
precisely to generalized linear methods for ODEs, a number of Runge--Kutta
methods, and Nordsieck methods. This probabilistic formulation of classic
methods is valuable in two ways: analytically, it highlights implicit prior
assumptions favoring certain approximate solutions to the IVP over others, and
gives a precise meaning to the old observation that these methods act like
filters. Practically, it endows the classic solvers with `docking points' for
notions of uncertainty and prior information about the initial value, the value
of the ODE itself, and the solution of the problem.