Bounds on the Posterior-to-Prior Ratios for Inclusion Belief under Bounded Differential Privacy
Abstract
Differential privacy has become the standard for generating privacy-protected data releases. However, differential privacy does not translate intuitively to disclosure risk. In particular, it remains unclear how much an adversary's belief about an individual's inclusion in a dataset can change after observing a protected release. To address this question, we derive upper and lower bounds on the posterior-to-prior ratios of inclusion beliefs under bounded probabilistic and approximate differential privacy. By assuming a worst-case adversary with all-but-one auxiliary information, i.e., knowledge of all except for one of the participants in a dataset, we obtain bounds that apply to any adversary. Because these bounds may fail with non-zero probability, we study the corresponding failure probability for the Gaussian mechanism. We derive a theoretical upper limit on this probability and compare it with Monte Carlo estimates across a wide range of parameter settings. The observed failure rate is several orders of magnitude smaller than its theoretical upper limit, indicating that the latter is highly conservative. These findings suggest that the inferential privacy guarantees provided by differentially private mechanisms may be substantially stronger in practice than what is implied by the theoretical upper limit.
Related
- Differential Privacy of Gaussian Process Posterior Sampling
- Lower bounds for Bayesian CVaR in interactive problems with local differential privacy
- On Inherent Privacy of Posterior Sampling: A Unified Rényi-Divergence Framework
- Prophet Inequalities under Local Differential Privacy
- On Privacy Mechanism Design Under a Bounded α-Lift Leakage Constraint with χ2-Utility
BibTeX
@misc{sorensen2026bounds,
title = {{Bounds on the Posterior-to-Prior Ratios for Inclusion Belief under Bounded Differential Privacy}},
author = {Jan Reiter Sørensen and Heidi Søgaard Christensen and Rasmus Rask Kragh Jørgensen and Martin Bøgsted},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30473},
eprint = {2608.30473},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30473},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bounds on the Posterior-to-Prior Ratios for Inclusion Belief under Bounded Differential Privacy
AU - Jan Reiter Sørensen
AU - Heidi Søgaard Christensen
AU - Rasmus Rask Kragh Jørgensen
AU - Martin Bøgsted
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30473
UR - https://arxiv.org/abs/2608.30473
AB - Differential privacy has become the standard for generating privacy-protected data releases. However, differential privacy does not translate intuitively to disclosure risk. In particular, it remains unclear how much an adversary's belief about an individual's inclusion in a dataset can change after observing a protected release. To address this question, we derive upper and lower bounds on the posterior-to-prior ratios of inclusion beliefs under bounded probabilistic and approximate differential privacy. By assuming a worst-case adversary with all-but-one auxiliary information, i.e., knowledge of all except for one of the participants in a dataset, we obtain bounds that apply to any adversary. Because these bounds may fail with non-zero probability, we study the corresponding failure probability for the Gaussian mechanism. We derive a theoretical upper limit on this probability and compare it with Monte Carlo estimates across a wide range of parameter settings. The observed failure rate is several orders of magnitude smaller than its theoretical upper limit, indicating that the latter is highly conservative. These findings suggest that the inferential privacy guarantees provided by differentially private mechanisms may be substantially stronger in practice than what is implied by the theoretical upper limit.
ER -