  We study a class of Gaussian processes for which the posterior mean, for a
particular choice of data, replicates a truncated Taylor expansion of any
order. The data consists of derivative evaluations at the expansion point and
the prior covariance kernel belongs to the class of Taylor kernels, which can
be written in a certain power series form. This permits statistical modelling
of the uncertainty in a variety of algorithms that exploit first and second
order Taylor expansions. To demonstrate the utility of this Gaussian process
model we introduce new probabilistic versions of the classical extended Kalman
filter for non-linear state estimation and the Euler method for solving
ordinary differential equations.
