On censored Markov chains, best augmentations and aggregation/disaggregation procedures

Consider the stationary distribution of a Markov chain censored to a subset of the original state space. It can be approximated by augmenting the substochastic matrix T representing transition probabilities in this subset into a stochastic matrix and solving for the corresponding stationary distribution. For the case of nearly uncoupled chains, under some mild technical assumptions, we find the best augmentation in the sense of minimizing the resulting error-bound, out of all augmentations which are based solely on the data given by T. Doing that for all submatrices representing transition probabilities inside all the subsets of a nearly uncoupled Markov chain constitutes part of the aggregation step in a standard aggregation/disaggregation procedure for approximating the stationary distribution of the original process.