If I have two random variables $X_n$ and $X_{n-1}$ where $X_n = \{{X_{n-1}, X_{n-1}-1\}} $ (can only have those two values). I also want to calculate the following expression:
$$P[X_{n+1} = i_n , X_{n} = i_n, X_{n-1} = i_{n-1},...] + P[X_{n+1} = i_n-1 , X_{n} = i_n, X_{n-1} = i_{n-1},...] \over P[X_{n} = i_n, X_{n-1} = i_{n-1},...] $$
Note that $X_n $ and $X_{n-1}$ have related value. Can I exploit this to make the above expression simpler?