Positivity of Renyi Mutual Information 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?
 A: EDIT. I justify the positivity of the Renyi mutual information using its interpretation as Renyi divergence. I follow the expositions in 
T. Cover, J. A. Thomas "Elements of Information Theory"  (chapter 2)
and
D. Xu, D. Erdogmuns  "Renyi's Entropy, Divergence and their Nonparametric Estimators"


*

*Shannon entropy and mutual information
In the setting of "classical" information theory the mutual information $I(X,Y)$ of the random variables $X$ and $Y$ is defined as 
$$I(X,Y):=D_{KL}(p_{XY}||p_Xq_Y),$$
where $D_{KL}(p_{XY}||p_Xq_Y),$ denotes the Kullback Leibler divergence (KL divergence) between the joint probability $p_{XY}$ and the product $p_Xq_Y$ of the prob. distribution of $X$ and $Y$. 
Using the Jensen inequality on the KL divergence it follows that $I(X,Y)$ is always non negative. I refer to the first reference for the computation in the discrete case.
Introducing the Shannon entropies $H(X)$ $H(Y)$ of $X$ resp. $Y$ and the conditional entropy $H(X|Y)$ we arrive at the equivalent formulation
$$I(X,Y)=H(X)+H(Y)-H(X|Y).$$ 


*

*Renyi Entropy and mutual information
Let us consider the Renyi $\alpha$-setting , now. With 
$$H_{\alpha}(X)=\frac{1}{1-\alpha}\log\int p^{\alpha}_X(x)dx$$
we denote the Renyi entropy of the r.v. $X$. The Renyi divergence of the distribution $g(x)$ from the distribution $f(x)$ is
$$D_{\alpha}(f||g):=\frac{1}{\alpha-1}\log\int f(x)\left(\frac{f(x)}{g(x)}\right)^{\alpha-1}dx.$$
It can be proved that (please see the second reference at pag.81)
$$D_{\alpha}(f||g)\geq 0 \forall ~f, g, \text{and}~\alpha>0,~~(*)$$
$$\lim_{\alpha\rightarrow 1}D_{\alpha}(f||g)=D_{1}(f||g)=D_{KL}(f||g).~~(*)$$
The Renyi mutual information $I_{\alpha}(X,Y)$ is defined naturally as the Renyi divergence between the joint distribution $p_{XY}$ of $X$ and $Y$ and the product of the marginal distributions $p_X$, $q_Y$, i.e.
$$I_{\alpha}(X,Y):=D_{\alpha}(p_{XY}||p_Xq_Y).$$
This is a definition; you can find it, for example, at pag. 83 in the second reference.
You can justify it through the overall $\alpha$-setting  and  the limit
$$\lim_{\alpha\rightarrow 1}I_{\alpha}(X,Y)=I(X,Y),$$
which follows from property $(**)$ of the Renyi divergence. This limit is parallel to the  fundamental $\lim_{\alpha\rightarrow 1}H_{\alpha}(X)=H(X):$
From property $(*)$ one derives nonnegativity of the Renyi mutual information. 
For these reasons, I would prove non negativity of the Renyi mutual information through the above definition. At the present stage I haven't been able to prove that
$$I_{\alpha}(X,Y)=H_{\alpha}(X)+H_{\alpha}(Y)-H_{\alpha}(X|Y),$$
or to find such characterization in the literature. Even in the discrete case I got blocked because of the coefficient $\frac{1}{1-\alpha}$ in front of the entropies. The cases $0<\alpha<1$ and $\alpha>1$ must be studied separately and it seems that a straightforward application of Jensen's inequality is not possible.
A: Unfortunately the quantity $I_q(X,Y) = H_q(X) + H_q(Y) - H_q(X,Y)$ is not always positive. This is particularly annoying given that this quantity has the useful property of being $0$ when $X$ and $Y$ are independent.
This is due to the fact that this definition implicitly states that the conditional Renyi entropy is:
$$H_q(X|Y) = H_q(X,Y) - H_q(Y),$$
and with this definition it is not always true that conditioning a random variable decreases its Renyi entropy. That is:
$$ H_q(X|Y) \leq H_q(X) \text{  is false}$$
From Conditional Renyi Entropies [Teixeira2012], we can find a joint probability distribution $(X,Y)$ for which the inequality above does not hold:
$$P(X = x_1, Y = y_1) = 0.2, P(X = x_2, Y = y_1) = 0.4$$
$$P(X = x_1, Y = y_2) = 0, P(X = x_2, Y = y_2) = 0.4$$
In this case,
$$I_q(X,Y) = \frac{1}{1-q} \log{\Big(\frac{\int \int p(x)^q p(y)^q}{\int \int p(x,y)^q}\Big)}$$
for $q = 3$ is equal to:
$$
\frac{1}{1-3}\log(\frac{(0.2 \cdot 0.4)^3 + (0.2 \cdot 0.6)^3 + (0.6\ \cdot 0.8)^3 + (0.4 \cdot 0.8)^3}{0.2^3 + 0.4^3 + 0.4^3}) = -\frac{1}{2} \log(\frac{0.1456}{0.136}) = - 0.015
$$
which is negative.
A: 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.
