# Bayesian Differential Privacy through Posterior Sampling

arXiv: https://arxiv.org/abs/1306.1066 · HTML: https://mlsanity.com/paper/1306.1066/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jun 2013.

## Built on by (most cited first, 18 of 18)

- [DP-PCA: Statistically Optimal and Differentially Private PCA](https://mlsanity.com/paper/2205.13709/) (2022)
- [Near Optimal Private and Robust Linear Regression](https://mlsanity.com/paper/2301.13273/) (2023)
- [Differentially Private Sampling from Distributions](https://mlsanity.com/paper/2211.08193/) (2022)
- [Non-parametric Differentially Private Confidence Intervals for the Median](https://mlsanity.com/paper/2106.10333/) (2021)
- [Insufficient Statistics Perturbation: Stable Estimators for Private Least Squares](https://mlsanity.com/paper/2404.15409/) (2024)
- [Privacy Amplification Via Bernoulli Sampling](https://mlsanity.com/paper/2105.10594/) (2021)
- [Mitigating Statistical Bias within Differentially Private Synthetic Data](https://mlsanity.com/paper/2108.10934/) (2021)
- [DP-Fast MH: Private, Fast, and Accurate Metropolis-Hastings for Large-Scale Bayesian Inference](https://mlsanity.com/paper/2303.06171/) (2023)
- [Differentially private partitioned variational inference](https://mlsanity.com/paper/2209.11595/) (2022)
- [Differentially private stochastic expectation propagation (DP-SEP)](https://mlsanity.com/paper/2111.13219/) (2021)
- [Differentially Private Statistical Inference through β-Divergence One Posterior Sampling](https://mlsanity.com/paper/2307.05194/) (2023)
- [Distributed Differentially Private Data Analytics via Secure Sketching](https://mlsanity.com/paper/2412.00497/) (2024)
- [Differentially Private Linear Regression and Synthetic Data Generation with Statistical Guarantees](https://mlsanity.com/paper/2510.16974/) (2025)
- [Robust and differentially private stochastic linear bandits](https://mlsanity.com/paper/2304.11741/) (2023)
- [The Normal Distributions Indistinguishability Spectrum and its Application to Privacy-Preserving Machine Learning](https://mlsanity.com/paper/2309.01243/) (2023)
- [DPpack: An R Package for Differentially Private Statistical Analysis and Machine Learning](https://mlsanity.com/paper/2309.10965/) (2023)
- [Data Analytics with Differential Privacy](https://mlsanity.com/paper/2311.16104/) (2023)
- [On Inherent Privacy of Posterior Sampling: A Unified R'enyi-Divergence Framework](https://mlsanity.com/paper/2610.19573/) (2026)

## Abstract (arXiv)

Differential privacy formalises privacy-preserving mechanisms that provide access to a database. We pose the question of whether Bayesian inference itself can be used directly to provide private access to data, with no modification. The answer is affirmative: under certain conditions on the prior, sampling from the posterior distribution can be used to achieve a desired level of privacy and utility. To do so, we generalise differential privacy to arbitrary dataset metrics, outcome spaces and distribution families. This allows us to also deal with non-i.i.d or non-tabular datasets. We prove bounds on the sensitivity of the posterior to the data, which gives a measure of robustness. We also show how to use posterior sampling to provide differentially private responses to queries, within a decision-theoretic framework. Finally, we provide bounds on the utility and on the distinguishability of datasets. The latter are complemented by a novel use of Le Cam's method to obtain lower bounds. All our general results hold for arbitrary database metrics, including those for the common definition of differential privacy. For specific choices of the metric, we give a number of examples satisfying our assumptions.