Renyi entropy of a random pair $(X,Y)$ with probability distribution $p_{X,Y}$ is defined by \begin{equation} H_\alpha(X,Y) = \frac{1}{1-\alpha}\log\sum_{x,y} p_{X,Y}(x,y)^\alpha. \end{equation} Assume that $X$ and $Y$ have countably infinite alphabets, and observe the case $0\leq\alpha\leq 1$.

It is known that this functional is subadditive, i.e., that $H_\alpha(X,Y)\leq H_\alpha(X) + H_\alpha(Y)$, only for $\alpha=0$ (Hartley entropy) and $\alpha=1$ (Shannon entropy).

My question is the following: Does there exist some upper bound on the joint entropy for $0<\alpha<1$, in terms of the marginal entropies? In fact, I don't even need an explicit upper bound, answering the following simpler question would suffice: If $H_\alpha(X)<\infty$ and $H_\alpha(Y)<\infty$, does there exist a constant $h$ such that $H_\alpha(X,Y)\leq h$?


1 Answer 1


The answer is no. Renyi entropy can even be unbounded and discontinuous over the set of probability distributions with given marginals (but not always, this depends on the marginals). You can find detailed proofs in http://arxiv.org/abs/1303.3235v1.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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