  Often the regression function appearing in fields like economics,
engineering, biomedical sciences obeys a system of higher order ordinary
differential equations (ODEs). The equations are usually not analytically
solvable. We are interested in inferring on the unknown parameters appearing in
the equations. Significant amount of work has been done on parameter estimation
in first order ODE models. Bhaumik and Ghosal (2014a) considered a two-step
Bayesian approach by putting a finite random series prior on the regression
function using B-spline basis. The posterior distribution of the parameter
vector is induced from that of the regression function. Although this approach
is computationally fast, the Bayes estimator is not asymptotically efficient.
Bhaumik and Ghosal (2014b) remedied this by directly considering the distance
between the function in the nonparametric model and a Runge-Kutta (RK$4$)
approximate solution of the ODE while inducing the posterior distribution on
the parameter. They also studied the direct Bayesian method obtained from the
approximate likelihood obtained by the RK4 method. In this paper we extend
these ideas for the higher order ODE model and establish Bernstein-von Mises
theorems for the posterior distribution of the parameter vector for each method
with $n^{-1/2}$ contraction rate.
