# Private Posterior distributions from Variational approximations

arXiv: https://arxiv.org/abs/1511.07896 · HTML: https://mlsanity.com/paper/1511.07896/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Nov 2015.

## Built on by (most cited first, 4 of 4)

- [Differentially Private Normalizing Flows for Density Estimation, Data Synthesis, and Variational Inference with Application to Electronic Health Records](https://mlsanity.com/paper/2302.05787/) (2023)
- [Simulation-based Bayesian Inference from Privacy Protected Data](https://mlsanity.com/paper/2310.12781/) (2023)
- [Fiducial Matching: Differentially Private Inference for Categorical Data](https://mlsanity.com/paper/2507.11762/) (2025)
- [One-Shot Private Confidence Regions via Resampling](https://mlsanity.com/paper/2610.08460/) (2026)

## Abstract (arXiv)

Privacy preserving mechanisms such as differential privacy inject additional randomness in the form of noise in the data, beyond the sampling mechanism. Ignoring this additional noise can lead to inaccurate and invalid inferences. In this paper, we incorporate the privacy mechanism explicitly into the likelihood function by treating the original data as missing, with an end goal of estimating posterior distributions over model parameters. This leads to a principled way of performing valid statistical inference using private data, however, the corresponding likelihoods are intractable. In this paper, we derive fast and accurate variational approximations to tackle such intractable likelihoods that arise due to privacy. We focus on estimating posterior distributions of parameters of the naive Bayes log-linear model, where the sufficient statistics of this model are shared using a differentially private interface. Using a simulation study, we show that the posterior approximations outperform the naive method of ignoring the noise addition mechanism.