Suppose that $X\rightarrow Y\rightarrow Z$ is a Markov chain with $X,Y$ and $Z$ are all Bernoulli variables, and we have the following assumptions:
- Between $X$ and $Y$: The joint distribution of $X$ and $Y$ are known, specifically $X=Y\oplus S_1$ for $S_1\sim \text{Bern}(p)$ and $X\sim \text{Bern}(a)$;
- Between $Y$ and $Z$: The conditional entropy $H(Y|Z)=C$.
Then for $X$ and $Z$, can we compute the conditional entropy $H(X|Z)$?