# 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.
• I voted up and go it as the answer as bayes theorem will solve the problem regardless indepence holds or not but It doesnt make sense, ok!, a finite markov chain will reset but index time not so $X_2$ is the first second transition ever, Im looking over if the results will be the same. Nov 20, 2021 at 22:19
• I misread What happened in the future variables clues what had happened in the past`, eng is not my main language, now past depending from future makes a little bit sense but thats another discussion thank you. Nov 20, 2021 at 22:49
• There is no way to get through this by total probability I was using the wrong theorem, in right way we would have $P(x_1)=\sum_{i \in S}P(x_1|x_0=i)P(x_0=i)$ how would I get $P(X_0=i)$? Nov 20, 2021 at 23:22