Problem Statement: Consider a graph $X_1\to X_2\to X_3\to X_4$ of binary random variables, and assume that the conditional probabilities between any two consecutive variables are given by \begin{align*} P(X_i=1|X_{i-1}=1)&=p\\ P(X_i=1|X_{i-1}=0)&=q\\ P(X_1=1)&=p_0. \end{align*} Compute the following probabilities \begin{align*} &P(X_1=1,X_2=0,X_3=1,X_4=0)\\ &P(X_4=1|X_1=1)\\ &P(X_1=1|X_4=1)\\ &P(X_3=1|X_1=0,X_4=1). \end{align*}
My Answer: First, we have \begin{align*} P(X_1=1,X_2=0,X_3=1,X_4=0) &=P(X_1=1)P(X_2=0|X_1=1)P(X_3=1|X_2=0)P(X_4=0|X_3=1)\\ &=p_0(1-p)^2q. \end{align*} This is due to the Rule of Product Decomposition. I understand how this works for conjunctions like the first probability. But I feel like that's a warmup question. Pearl has no examples to show how to compute these probabilities when you leave out terms in the graph. Can you please give me some hints on the last three probabilities? For example, how do you work forward through a graph, such as for $P(X_4=1|X_1=1),$ versus working backwards through the graph, such as for $P(X_1=1|X_4=1)?$
Thanks for your time!