The work here explores new numerical methods for supporting a Bayesian approach to parameter estimation of dynamic systems. This is primarily motivated by the goal of providing accurate quantification of estimation error that is valid for arbitrary, and hence even very short length data records. The main innovation is the employment of the Metropolis--Hastings algorithm to construct an ergodic Markov chain with invariant density equal to the required posterior density. Monte--Carlo analysis of samples from this chain then provide a means for efficiently and accurately computing posteriors for model parameters and arbitrary functions of them.