A basic doubt on Markov Chain / Conditioning

Question

Suppose we want to calculate $n$-step transition probability of a Markov chain conditioned on the fact that it does not pass through some particular state. Can I do this by removing that state from the Markov chain (after appropriate transfer of probability) and then calculating transition probability in this new matrix ?

Answer 1

The problem is that "appropriate transfer of probability" is non trivial. Consider the markov chain with three states, $A,B,C$ such that $P(A \to B) = .2$, $P(A \to A) = .8$, $P(B \to B) = .1$, $P(B \to B) = .1$, $P(B \to C) = .9$, $P(C \to C) = .1$, and $P(C \to B) = .9$ with all other probabilities zero.

If you want to compute, say, the 4-step transition probability of this chain conditioned on not passing through C, it is not enough to simply recompute the transition probabilities starting at $B$ to remove $C$, since conditioning means is now less likely for any path to have gone through $B$ also.