I have a Markov chain with a state space $S$ and its transition probability matrix generated mathematically as $P$. I have created another Markov chain with a reduced state space $S_r$ made by partitioning $S$. The transition of this reduced state space Markov chain is mathematically generated as $P_r$.
Using the ordinary lumpability described in [1], I have determined the matrix $V$ and $U$. The Markov chain is not meeting the conditions for lumpability in Theorem 6.3.4 if [1], since I'm getting $$ VUPV \ne PV.$$ However, I'm still getting the transition probability of the reduced Markov chain equal to the method to calculate transition probability matrix using the lumpability method, i.e $$P_r = UPV.$$
Currently I'm able to do this numerically, over multiple small examples and its been consistently giving the above results of $P_r = UPV$, despite $VUPV \ne PV$. I don't have a mathematical proof yet on why this result exists.
My questions are:
- Does my reduced Markov chain still represent the original Markov chain because $P_r = UPV$, simply because I have some special structure to the original Markov chain?
- Are there any literature references that allow for my result?
[1] Kemeny, J. G., & Snell, J. L. (1960). Finite Markov Chains. Van Nostrand.
