  An ordinary differential equation (ODE) model, whose regression curves are a
set of solution curves for some ODEs, poses a challenge in parameter
estimation. The challenge, due to the frequent absence of analytic solutions
and the complicated likelihood surface, tends to be more severe especially for
larger models with many parameters and variables. Yang and Lee (2021) proposed
a state-space model with variational Bayes (SSVB) for ODE, capable of fast and
stable estimation in somewhat large ODE models. The method has shown excellent
performance in parameter estimation but has a weakness of underestimation of
the posterior covariance, which originates from the mean-field variational
method. This paper proposes a way to overcome the weakness by using the Laplace
approximation. In numerical experiments, the covariance modified by the Laplace
approximation showed a high degree of improvement when checked against the
covariances obtained by a standard Markov chain Monte Carlo method. With the
improved covariance estimation, the SSVB renders fairly accurate posterior
approximations.
