TITLE: Positivity of Renyi Mutual Information
QUESTION [7 upvotes]: The differential Renyi entropy for a probability distribution is given by $H_q(P(X))=\frac{1}{1-q}\log\int p^q(x)dx$. In the limit of $q\to 1$, it reduces to the usual Shannon entropy. We can write down the mutual information between two variables X and Y simply by $I(X;Y)=H_q(P(X))+H_q(P(Y))-H_q(P(X,Y))$. Is this always a non-negative quantity? Again, in the case $q=1$ it is very easy to show it, but what about in general?

REPLY [2 votes]: The following paper elaborates on a few different possible definitions of "Rényi mutual information". It points out the negativity issue raised on Simone's answer and observes the nonnegativity of the proposal in Avitus' answer. 
α-Mutual Information
Sergio Verdú
http://www.ita.ucsd.edu/workshop/15/files/paper/paper_374.pdf
Verdú comes down in favor of a slight variant of Avitus' definition (attributed to Sibson):
$$I_\alpha(X;Y) = \min_{q_Y} D_\alpha(p_{XY} \| p_X q_Y)$$
where $p_X$ is the marginal of $X$ under $p_XY$ and $q_Y$ ranges over all possible distributions for $Y$ (not necessarily the marginal corresponding to $p_{XY}$). 
Perhaps the simplest takeaway here is that there is no single clear "Rényi mutual information". Which version you need will vary based on context.