  Coarse-graining techniques play a central role in reducing the complexity of
stochastic models, and are typically characterised by a mapping which projects
the full state of the system onto a smaller set of variables which captures the
essential features of the system. Starting with a continuous-time Markov chain,
in this work we propose and analyse an effective dynamics, which approximates
the dynamical information in the coarse-grained chain. Without assuming
explicit scale-separation, we provide sufficient conditions under which this
effective dynamics stays close to the original system and provide quantitative
bounds on the approximation error. We also compare the effective dynamics and
corresponding error bounds to the averaging literature on Markov chains which
involve explicit scale-separation. We demonstrate our findings on an
illustrative test example.
