\esssup_{P_{{\boldsymbol{Z}}}} \frac{\dv P_{W\! {\boldsymbol{Z}}}}{\dv P_W\!P_{{\boldsymbol{Z}}}}.$$ Applying Markov’s inequality with respect to $P_W$ to the random variable $\dv P_{W{\boldsymbol{Z}}} /\dv P_W \!P_{{\boldsymbol{Z}}}$, we conclude that, with probability at least $1-\delta$ under $P_{W}$, $$\label{eq:cor_uncond_sd_leakage_proof_infdens_bound}
\imath(W,{\boldsymbol{Z}}) \leq \log \esssup_{P_{{\boldsymbol{Z}}} } \Exop_{P_{W}}\lefto[  \frac{\dv P_{W\! {\boldsymbol{Z}}}}{\dv P_W\!P_{{\boldsymbol{Z}}}}\right] + \log\frac{1}{\delta}.$$ By using the union bound to combine  with  and performing the substitution $\delta\rightarrow \delta/2$, we obtain .

To prove , we first apply Markov’s inequality and then perform a change of measure to conclude that the following inequalities hold with probability at least $1-\delta$ under $P_{W\! {\boldsymbol{Z}}}$: $$\label{eq:cor_singledraw_uncond_alphadiv_pf_markov_com}
\imath(W,{\boldsymbol{Z}}) \leq \log\Exop_{P_{W\!{\boldsymbol{Z}}}}\lefto[\frac{\dv P_{W\! {\boldsymbol{Z}}}}{\dv P_W\!P_{{\boldsymbol{Z}}}}\right] + \log\frac{1}{\delta} \leq  \log\Exop_{P_W\!P_{{\boldsymbol{Z}}}}\lefto[\left(\frac{\dv P_{W\! {\boldsymbol{Z}}}}{\dv P_W\!P_{{\boldsymbol{Z}}}}\right)^2\right] + \log\frac{1}{\delta}.$$ Next, we apply Hölder’s inequality twice as follows. Let $\alpha,\gamma,\alpha',\gamma'> 1$ be constants such that $1/\alpha+1/\gamma=1/\alpha'+1/\gamma'=1$. Then,

[rCl]{} \_[P\_WP\_]{} && \_[P\_W]{}\
&& \_[P\_W]{}\^[1/’]{}\_[P\_W]{}\^[1/’]{}.

Setting $\alpha=\alpha'$, which implies $\gamma=\gamma'$, we conclude that

[rCl]{} \_[P\_WP\_]{} && \_[P\_WP\_ ]{}\^[1/]{}+ \_[P\_WP\_ ]{}\^[1/ ]{}\[eq:choosePandQ\]\
\[eq:cor\_uncond\_sd\_alphadiv\_proof\_infdens\_bound\] &=&D\_(P\_[W ]{}||P\_WP\_)+D\_(P\_[W ]{}||P\_WP\_).

Substituting  into , and then combining  with  through the union bound, we establish  after the substitution $\delta\rightarrow \delta/2$.

The bound in  can be relaxed to recover the maximal leakage bound in [@esposito19-12a Cor. 10], the max information bound in [@dwork15-06a Thm. 4] for the case in which the parameter $\beta$ 