In the converse proof in information theory, using Fano's inequality, at the end we would have a term like

$I(X^n;Y^n)\leq nI(X;Y)$

I was wondering can we prove something like this for relative entropy? Something like this?

$D(P_{X^n}||Q_{X^n})\leq n D(P_X||Q_X)$


No, I think you cannot. Take this counterexample:

Let $P_{X^n}$ be the joint distribution of $n$ consecutive samples of a stationary Gaussian process, where the stationary distribution is $P_X$. Let $Q_{X^n}=\prod_{i=1}^nP_X$ and $Q_X=P_X$. It follows that $D(P_X\Vert Q_X)=0$, while $D(P_{X^n}\Vert Q_{X^n})$ need not vanish.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.