We present a simulation methodology for Bayesian estimation of rate
parameters in Markov jump processes arising for example in stochastic kinetic
models. To handle the problem of missing components and measurement errors in
observed data, we embed the Markov jump process into the framework of a general
state space model. We do not use diffusion approximations. Markov chain Monte
Carlo and particle filter type algorithms are introduced, which allow sampling
from the posterior distribution of the rate parameters and the Markov jump
process also in data-poor scenarios. The algorithms are illustrated by applying
them to rate estimation in a model for prokaryotic auto-regulation and in the
stochastic Oregonator, respectively.