Asymptotic behavior of the empirical failure probability

Characterize the asymptotic behavior of the empirical failure probability \(\overline{F}\) as the privacy parameter \(\varepsilon\) tends to infinity, determining whether it converges to a strictly positive limit or exhibits qualitatively different behavior when the increasing variability of the posterior-to-prior ratio outweighs the widening admissible interval.

Background

The paper studies the probability that posterior-to-prior inclusion-belief ratio bounds fail under the Gaussian mechanism. In the parameter sweep over ε∈[0.2,2.0]\varepsilon\in[0.2,2.0], the empirical failure probability F‾\overline{F} decreases as ε\varepsilon increases, while the admissible interval for the ratio widens.

The authors emphasize that the simulations cover only a finite range of ε\varepsilon and therefore do not resolve the behavior as ε→∞\varepsilon\to\infty. The unresolved issue is whether the widening bounds eventually dominate the increasing variability induced by weaker privacy, or whether the failure probability remains bounded away from zero or changes qualitatively.

References

Although the simulations cover only a finite range of $\varepsilon$ and therefore do not determine this asymptotic behaviour, we conjecture that $\overline{F}$ may converge to a strictly positive limit or eventually exhibit qualitatively different behaviour if the increasing variability of $\mathscr{R}$ outweighs the widening of the admissible interval.

— Bounds on the Posterior-to-Prior Ratios for Inclusion Belief under Bounded Differential Privacy  (2608.30473 - Sørensen et al., 31 Aug 2026) in Section 6, subsection “Results”