limit $\alpha\rightarrow 1$, it reduces to the relative entropy $\relent{P_{W\!{\boldsymbol{Z}}}}{P_W\!P_{{\boldsymbol{Z}}}}$. The conditional Rényi divergence of order $\alpha$ is given by [@verdu15-02a] $$\label{eq:def_cond_renyi_divergence}
\alphaconrelent{\alpha}{P_{W\vert {\widetilde{\boldsymbol{Z}}}\!{\boldsymbol{S}}}P_{{\boldsymbol{S}}\vert {\widetilde{\boldsymbol{Z}}}} }{P_{W\vert {\widetilde{\boldsymbol{Z}}}}P_{{\boldsymbol{S}}\vert {\widetilde{\boldsymbol{Z}}}}}{P_{{\widetilde{\boldsymbol{Z}}}}} = \frac{1}{\alpha-1}\log \Exop_{P_ {{\widetilde{\boldsymbol{Z}}}}P_{W\vert {\widetilde{\boldsymbol{Z}}}}P_{{\boldsymbol{S}}\vert {\widetilde{\boldsymbol{Z}}}}}\lefto[\exp\lefto(\alpha{{\imath}(W,{\boldsymbol{S}}\vert {\widetilde{\boldsymbol{Z}}})}\right) \right].$$ The $\alpha$-mutual information, which is studied in depth in [@verdu15-02a], is defined as$$\label{eq:def_alpha_MI}
I_\alpha({\boldsymbol{Z}};W) = \frac{1}{\alpha-1}\log \Exop_{P_W}^\alpha\lefto[ \Exop_{P_{{\boldsymbol{Z}}} }^{1/\alpha} \lefto[\exp\lefto(\alpha{{\imath}(W,{\boldsymbol{Z}})}\right)  \right] \right].$$ In the limit $\alpha\rightarrow 1$, it reduces to the mutual information $I(W;{\boldsymbol{Z}})$, whereas for $\alpha\rightarrow \infty$, it becomes the maximal leakage [@issa16-a]: $$\label{eq:def_maximal_leakage}
\mathcal{L}({\boldsymbol{Z}}\rightarrow W)=\log \Exop_{P_W}\lefto[ \esssup_{P_{{\boldsymbol{Z}}}}\exp\lefto({{\imath}(W,{\boldsymbol{Z}})}\right)  \right].$$ The conditional  $\alpha$-mutual information does not have a commonly accepted definition. In [@Tomamichel2018], three definitions are provided and given operational interpretations, two of which have known closed-form expressions. The first coincides with the conditional Rényi divergence, while the second, which we will term $I_\alpha(W;{\boldsymbol{S}}\vert {\widetilde{\boldsymbol{Z}}})$, is defined as $$\label{eq:def_cond_alpha_MI}
I_\alpha(W; \boldsymbol{S}\vert  \tilde{\boldsymbol{Z}} )=\frac{1}{\alpha-1}\log \Exop_{P_ {\tilde{\boldsymbol{Z}}}}\lefto[\Exop_{P_{W\vert \tilde{\boldsymbol{Z}} } }^{\alpha}\lefto[ \Exop_{P_{ \boldsymbol{S}\vert \tilde{\boldsymbol{Z}}}}^{1/\alpha}
\lefto[\exp\lefto(\alpha{{\imath}(W,{\boldsymbol{S}}\vert {\widetilde{\boldsymbol{Z}}})}\right) \right]\right]\right].$$ In the limit $\alpha\rightarrow \infty$, this reduces to the conditional maximal leakage[@issa16-a Thm. 6] $$\label{eq:def_cond_maximal_leakage}
 \mathcal{L}({\boldsymbol{S}}\rightarrow W\vert {\widetilde{\boldsymbol{Z}}})=\log\esssup_{P_{{\widetilde{\boldsymbol{Z}}}}} \Exop_{P_{W\vert {\widetilde{\boldsymbol{Z}}}}}\lefto[ \esssup_{P_{{\boldsymbol{S}}\vert {\widetilde{\boldsymbol{Z}}}} }\exp\lefto({{\imath}(W,{\boldsymbol{S}}\vert {\widetilde{\boldsymbol{Z}}})}\right) \right].$$ Note that ${\boldsymbol{S}}$ and ${\widetilde{\boldsymbol{Z}}}$ are independent in the random-subset setting. Hence, $P_{{\boldsymbol{S}}\vert {\widetilde{\boldsymbol{Z}}}}$ can be replaced by $P_{{\boldsymbol{S}}}$ in , , and .

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