$Y$ is a function of $X$: making an inference based on the markovity of $ X$

Question

In the information theory book by Cover and Thomas it is written:

if $X$ is markov and $Y$ is a function of $X$ then:

$H(Y_n|Y_{n-1},Y_{n-2},...,Y_1,X_1)=H(Y_n|Y_{n-1},Y_{n-2},...,Y_1,X_1,X_0,X_{-1},...,X_{-k})$ because $X$ is markov.

Can someone please explain how they made this inference? It's not very clear to me.

Thank you!!

Answer 1

The expression should follow from the property of Markov process. For a Markov process X(t) (with t being a time-index) the future states don't depend on the past states when the present state is given. In other words, knowledge of history of the random process doesn't add any new information.

So, for X is Markov and Y is a function of X, adding historical information (conditionally) to the function H, that is, having X(t) only for t=1 conditionally in the function H is the same as having X(t) for t = 1, 0, -1, -2, ..., -k conditionally. What matters here is the X1 in the condition. Hope this makes sense.