  Variational approaches to data assimilation, and weakly constrained four
dimensional variation (WC-4DVar) in particular, are important in the
geosciences but also in other communities (often under different names). The
cost functions and the resulting optimal trajectories may have a probabilistic
interpretation, for instance by linking data assimilation with Maximum
Aposteriori (MAP) estimation. This is possible in particular if the unknown
trajectory is modelled as the solution of a stochastic differential equation
(SDE), as is increasingly the case in weather forecasting and climate
modelling. In this case, the MAP estimator (or "most probable path" of the SDE)
is obtained by minimising the Onsager--Machlup functional. Although this fact
is well known, there seems to be some confusion in the literature, with the
energy (or "least squares") functional sometimes been claimed to yield the most
probable path. The first aim of this paper is to address this confusion and
show that the energy functional does not, in general, provide the most probable
path. The second aim is to discuss the implications in practice. Although the
mentioned results pertain to stochastic models in continuous time, they do have
consequences in practice where SDE's are approximated by discrete time schemes.
It turns out that using an approximation to the SDE and calculating its most
probable path does not necessarily yield a good approximation to the most
probable path of the SDE proper. This suggest that even in discrete time, a
version of the Onsager--Machlup functional should be used, rather than the
energy functional, at least if the solution is to be interpreted as a MAP
estimator.
