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A stochastic matrix with states $S_1$, $S_2$, $S_3$, $S_4$ is given, now we would like to build up another stochastic matrix with finer states, meaning that the states $S_1$ will be considered as $S_1=Z_1+Z_2$ ('$+$' sign stands for the union of two new states), and $S_2=Z_3+Z_4$, $S_3=Z_5+Z_6$, and $S_4=Z_7+Z_8$. Indeed, this kind of information could be available from Bayesian Statistics. I am just wondering if anybody studied or knows references for such kind of problem.

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One could duplicate each $S$ state hence the construction you have in mind seems to be always trivially possible--unless you have more constraints in mind, in which case you should explain them. An interesting concept is the opposite one, where one groups some states of a Markov chain and one considers the resulting process, called the lumped chain.

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