# Differentially Private Bayesian Linear Regression

arXiv: https://arxiv.org/abs/1910.13153 · HTML: https://mlsanity.com/paper/1910.13153/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2019.

## Built on by (most cited first, 24 of 24)

- [Differentially Private Regression with Unbounded Covariates](https://mlsanity.com/paper/2202.11199/) (2022)
- [Incentives in Private Collaborative Machine Learning](https://mlsanity.com/paper/2404.01676/) (2024)
- [Noise-Aware Statistical Inference with Differentially Private Synthetic Data](https://mlsanity.com/paper/2205.14485/) (2022)
- [Non-parametric Differentially Private Confidence Intervals for the Median](https://mlsanity.com/paper/2106.10333/) (2021)
- [Differentially Private Bayesian Neural Networks on Accuracy, Privacy and Reliability](https://mlsanity.com/paper/2107.08461/) (2021)
- [Private Regression via Data-Dependent Sufficient Statistic Perturbation](https://mlsanity.com/paper/2405.15002/) (2024)
- [Gaussian Processes with Differential Privacy](https://mlsanity.com/paper/2106.00474/) (2021)
- [Differentially Private Nonparametric Regression Under a Growth Condition](https://mlsanity.com/paper/2111.12786/) (2021)
- [Sharper Utility Bounds for Differentially Private Models](https://mlsanity.com/paper/2204.10536/) (2022)
- [DPVIm: Differentially Private Variational Inference Improved](https://mlsanity.com/paper/2210.15961/) (2022)
- [Is merging worth it? Securely evaluating the information gain for causal dataset acquisition](https://mlsanity.com/paper/2409.07215/) (2024)
- [Locally Differentially Private Bayesian Inference](https://mlsanity.com/paper/2110.14426/) (2021)
- [Differentially Private Distributed Bayesian Linear Regression with MCMC](https://mlsanity.com/paper/2301.13778/) (2023)
- [Differentially Private Statistical Inference through β-Divergence One Posterior Sampling](https://mlsanity.com/paper/2307.05194/) (2023)
- [Simulation-based Bayesian Inference from Privacy Protected Data](https://mlsanity.com/paper/2310.12781/) (2023)
- [Differentially private Bayesian tests](https://mlsanity.com/paper/2401.15502/) (2024)
- [Fiducial Matching: Differentially Private Inference for Categorical Data](https://mlsanity.com/paper/2507.11762/) (2025)
- [Computation-Utility-Privacy Tradeoffs in Bayesian Estimation](https://mlsanity.com/paper/2603.18254/) (2026)
- [The Normal Distributions Indistinguishability Spectrum and its Application to Privacy-Preserving Machine Learning](https://mlsanity.com/paper/2309.01243/) (2023)
- [Noise-Aware Differentially Private Variational Inference](https://mlsanity.com/paper/2410.19371/) (2024)
- [Optimal Survey Design for Private Mean Estimation](https://mlsanity.com/paper/2501.18121/) (2025)
- [On Reconstructing Training Data From Bayesian Posteriors and Trained Models](https://mlsanity.com/paper/2507.18372/) (2025)
- [Differentially Private Two-Stage Gradient Descent for Instrumental Variable Regression](https://mlsanity.com/paper/2509.22794/) (2025)
- [Differentially Private Truncation of Unbounded Data via Public Second Moments](https://mlsanity.com/paper/2602.22282/) (2026)

## Abstract (arXiv)

Linear regression is an important tool across many fields that work with sensitive human-sourced data. Significant prior work has focused on producing differentially private point estimates, which provide a privacy guarantee to individuals while still allowing modelers to draw insights from data by estimating regression coefficients. We investigate the problem of Bayesian linear regression, with the goal of computing posterior distributions that correctly quantify uncertainty given privately released statistics. We show that a naive approach that ignores the noise injected by the privacy mechanism does a poor job in realistic data settings. We then develop noise-aware methods that perform inference over the privacy mechanism and produce correct posteriors across a wide range of scenarios.