Forgive me for my weak statistic background, hopefully what I'm asking makes sense. So some quick background, I have one markov chain from a data set and many additional chains that I'm producing from a simulation. I'm looking for a means of comparing these chains, to produce some measure of accuracy in my simulation.
I know the MLE of the transition matrix of one chain is simply $p_{ij} = n_{ij}/\sum_{i,j=1}^{n}n_{ij}$ where $p_{ij}$ is the probability from transitioning from $i$ to $j$ and $n_{ij}$ is the number of transitions. Is there some way of estimating the likelihood that two chains are derived from same underlying transition matrix? If the MLE of the transition matrix of simulation data is generated can these transition matrices be compared?
thanks for any help!