  It has recently been established that the numerical solution of ordinary
differential equations can be posed as a nonlinear Bayesian inference problem,
which can be approximately solved via Gaussian filtering and smoothing,
whenever a Gauss--Markov prior is used. In this paper the class of $\nu$ times
differentiable linear time invariant Gauss--Markov priors is considered. A
taxonomy of Gaussian estimators is established, with the maximum a posteriori
estimate at the top of the hierarchy, which can be computed with the iterated
extended Kalman smoother. The remaining three classes are termed explicit,
semi-implicit, and implicit, which are in similarity with the classical notions
corresponding to conditions on the vector field, under which the filter update
produces a local maximum a posteriori estimate. The maximum a posteriori
estimate corresponds to an optimal interpolant in the reproducing Hilbert space
associated with the prior, which in the present case is equivalent to a Sobolev
space of smoothness $\nu+1$. Consequently, using methods from scattered data
approximation and nonlinear analysis in Sobolev spaces, it is shown that the
maximum a posteriori estimate converges to the true solution at a polynomial
rate in the fill-distance (maximum step size) subject to mild conditions on the
vector field. The methodology developed provides a novel and more natural
approach to study the convergence of these estimators than classical methods of
convergence analysis. The methods and theoretical results are demonstrated in
numerical examples.
