Markov subchain

Question

If $X \rightarrow (Y_1,Y_2) \rightarrow Z$ is a Markov chain, does this imply that $X \rightarrow Y_1 \rightarrow Z$ is also a Markov chain?

Answer 1

In general, no. Here is an example. Consider the Markov Chain $X\rightarrow (Y_1,Y_2)\rightarrow Z$, with $Z=Y_1+Y_2$. Now, $X\rightarrow Y_1 \rightarrow Z$ holds if and only if $Z$ and $X$ are independent when conditioned on $Y_1$, i.e., $p(z,x|y_1)=p(z|y_1) p(x|z_1)$. Clearly, conditioned on $Y_1=y_1$, the value of $Z=y_1+Y_2$ depends on $Y_2$. If $Y_2$ is not independent of $X$ given $Y_1$, this means that $Z$ is also not independent of $X$ given $Y_1$ and $X\rightarrow Y_1 \rightarrow Z$ does not hold.