Factorising and drawing Markov chains Problem

Question

My question for today is:

Let $X_1, X_2, X_3$ be a Markov chain, with the subscript denoting the time index. Give a factorisation of the joint distribution $p(X_1, X_2, X_3)$ that is simplified by the Markov property and draw a graphical model representing this factorisation.

Ive graphically drawn plenty of Markov chains before but im not too sure about this question ive been given so if anyone can help me out that would be fantastic

Answer 1

Markov chain means given the present, the future is independent from the past.

Given $X_2$, $X_1$ and $X_3$ are independent.

\begin{align} p(X_1,X_2,X_3) &= p(X_1)p(X_2|X_1)p(X_3|X_1,X_2) \\ &= p(X_1)p(X_2|X_1)p(X_3|X_2) \end{align}

Notice that $$p(X_3|X_1,X_2) = p(X_3|X_2)$$

due to Markov property.