Evaluating $P(X_{3}=j,X_{1}=i|X_{2}=k)$ on markov chain trouble I just would like to evaluate $P(X_{3}=j,X_{1}=i|X_{2}=k)$
So I tried:
$P(X_{3}=j,X_{1}=i|X_{2}=k)
=P(X_{3}=j|X_{2}=k)P(X_{1}=i|X_{2}=k)$
The trouble is evaluating $P(X_{1}=i|X_{2}=k)$
I tried $P(X_{1}=i|X_{2}=k)=P(X_{1}=i)$ since I think the past doesnt depend from future and I applied total probability theorem so:
$P(X_1=i)=\sum_{l \in S}P(X_1=i|X_0=l)$ where $S$ is the state space.
But I'm not sure if this independence holds if not what should I do?
PS: Assume transition matrix is known so $\sum_{l \in S}P(X_1=i|X_0=l)$ is known.

What happened in the future variables clues what had happened in the past, so the assumption $p(x_1|x_2)=p(x_1)$ is not correct. You should use Bayes theorem to calculate it:
$$p(x_1|x_2)=\frac{p(x_2|x_1)p(x_1)}{p(x_2)}$$
You can calculate $p(x_i)$ from total probability theorem, or alternatively do the same calculations using some matrix multiplications as described in this notes, page 10.
