The following problem is giving me a bit of a headache:
Let $X, Z$ be a pair of random variables.
Under which conditions can I construct a random variable $Y$ such that
$X, Y, Z$ is a Markov chain? In order to avoid trivial solutions such as $X=Y$ I additionally require the construction to be symmetrical with respect to $X$ and $Z$.
That is, if I change $X$ and $Z$ when defining $Y$, I obtain the same random variable.
I am thinking of $Y$ of some sort of variable that captures the common “information” between $X$ and $Z$. Maybe it even holds that $I(X, Y) = I(Y, Z)$?
So far I could only find such $Y$ if I assume $X$ and $Y$ to be correlated standard normal distributions. An answer under any (non totally trivial) assumption is welcome 🙂