Modeling and inferring stochastic biomolecular processes based on single-cell data requires an extension of the traditional Markov chain description to account for the random molecular environment into which the process of interest is embedded. In particular, we seek an isolated process model that behaves as if the process was still embedded into the molecular environment. Based on that novel process model we develop a Bayesian inference framework that resorts to traditional MCMC schemes in combination with sequential Monte Carlo techniques. We apply the framework to live-cell imaging data of a inducible gene expression system in budding yeast and show that it allows to separate intrinsic from extrinsic noise components from single measurements without the need for dedicated dual-color constructs.