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