I am looking for advice on the following practical problem. Please keep in mind that this came up in a practical application.

In the context of Markov chains, we have $N$ states, with $N$ very large. (In this application $N$ is on the order of $10^{100000}$ or so.) Every state is reachable from every other state (possibly in more than one step).

We have two transition probability matrices, $A$ and $B$.

I can *probably* calculate $A_{ij}$ for any two states $i$ and $j$ if needed, but what I can do *easily* is just propagate the states (simulate the process).

Suppose we start from some arbitrary state and let the system evolve for a very long time, obtaining a state $k$.

Given this state $k$, what is the probability that the system has been evolved using $A$? What is the probability that it was evolved using $B$?

In other words: I can easily simulate both processes on a computer. Given some state $k$, are there practical techniques to find out which of the two processes is more likely to have produced it?

*Note:* I think this Markov chain is reversible.

_{(I expect that the question will probably need clarifications once I manage to understand it more deeply. Please help with this.)}