  When taking the model error into account in data assimilation, one needs to
evaluate the prior distribution represented by the Onsager--Machlup functional.
Through numerical experiments, this study clarifies how the prior distribution
should be incorporated into cost functions for discrete-time estimation
problems. Consistent with previous theoretical studies, the divergence of the
drift term is essential in weak-constraint 4D-Var (w4D-Var), but it is not nec
essary in Markov chain Monte Carlo with the Euler scheme. Although the former
property may cause difficulties when implementing w4D-Var in large systems,
this paper proposes a new technique for estimating the divergence term and its
derivative.
