An efficient sampling method is proposed to solve the stochastic optimal control problem in the context of data assimilation for the estimation of a random parameter. It is based on Bayesian inference and the Markov Chain Monte Carlo technique, which exploits the relation between the inverse Hessian of the cost function and the error covariance matrix to accelerate convergence of the sampling method. The efficiency and accuracy of the method is demonstrated in the case of the optimal control problem governed by the nonlinear Burgers equation with a viscosity parameter that is a random field.