Large-Sample Bayesian Approximations for Privatized Data

Jordan Awan; Roberto Molinari; Xi Chen

arxiv: 2604.24817 · v1 · submitted 2026-04-27 · 📊 stat.ME · math.ST· stat.AP· stat.TH

Large-Sample Bayesian Approximations for Privatized Data

Jordan Awan , Xi Chen , Roberto Molinari This is my paper

Pith reviewed 2026-05-08 02:11 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.APstat.TH

keywords differential privacyBayesian inferenceprivatized dataasymptotic validitydata imputationfrequentist propertiesAmerican Community Surveystatistical inference

0 comments

The pith

A two-step approximate Bayesian sampler for privatized data is asymptotically valid under mild assumptions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a practical method for Bayesian statistical inference on data protected by differential privacy. The approach imputes the confidential data and then draws samples from the posterior as if no privacy noise had been added. The authors establish that this approximation is asymptotically valid for large samples under mild assumptions on the privacy mechanism and data process. Simulations further show that the method delivers conservative frequentist properties, making it useful for analyzing large real-world privatized datasets such as the American Community Survey.

Core claim

The central claim is that the approximate Bayesian sampler, obtained by first imputing the confidential data from the privatized observations and then sampling from the non-private posterior, is asymptotically valid as the sample size grows large. This validity holds under mild assumptions concerning the differential privacy mechanism, the imputation step, and the data-generating process. The method is demonstrated to work in simulated settings and applied to an analysis of homeownership drivers using the 2022 American Community Survey, while simulations confirm conservative frequentist behavior.

What carries the argument

The two-step procedure of imputing the confidential data from the privatized version followed by sampling from the non-private posterior distribution.

If this is right

Scales Bayesian inference to large privatized datasets without requiring strong parametric assumptions.
Delivers conservative frequentist coverage properties in addition to asymptotic Bayesian validity.
Facilitates analysis of real data products such as the American Community Survey on topics like homeownership.
Addresses scalability issues in existing methods for statistical inference on privatized data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This approach may extend to other data-sharing mechanisms that permit imputation of original values.
Large sample sizes could make the impact of privacy noise negligible when imputation is reliable.
Further validation could involve applying the method to additional privatized government datasets.
The conservative nature might be advantageous in applications where avoiding overconfidence is critical.

Load-bearing premise

The mild assumptions on the differential privacy mechanism, the imputation procedure, and the underlying data-generating process must hold in the specific application.

What would settle it

A simulation with increasing sample sizes in which the imputation inaccurately recovers the confidential data distribution or the privacy mechanism deviates from the assumed conditions, causing the approximate posterior to fail to concentrate at the true parameter or to lose frequentist coverage.

read the original abstract

The increased use of differential privacy (DP) has allowed the sharing of large amounts of data while reducing the risk of disclosure of sensitive information at the individual level. However, the noise introduced by DP methods makes performing statistical inference more challenging. While various methods have been proposed to address different inferential tasks, they often require strong parametric assumptions and/or do not scale well with sample sizes (e.g. U.S. Census products). In response to these limitations, we propose an approximate Bayesian method to analyze privatized data products, which uses a two-step approach of imputing the confidential data and then sampling from the non-private posterior, and which is inspired by the method of Guha and Reiter (2025). We prove that this approximate sampler is asymptotically valid under mild assumptions. While this approach is motivated by Bayesian theory, we show through simulations that it provides conservative frequentist properties as well. We demonstrate the utility of our method by applying it in simulated settings as well as for an analysis on the drivers of homeownership via the 2022 American Community Survey.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper offers a practical two-step Bayesian approximation for inference on large differentially private datasets, backed by an asymptotic proof and simulation checks.

read the letter

This paper's main contribution is a two-step approximate Bayesian method for analyzing data that has been privatized with differential privacy. You impute the confidential data first, then sample from the non-private posterior as if nothing had been privatized. They prove this sampler is asymptotically valid under mild assumptions, and the simulations indicate it gives conservative frequentist coverage as well. The approach is tested on simulated data and applied to the 2022 American Community Survey to look at drivers of homeownership. What works here is the practicality for large datasets. Many existing ways to handle inference under DP either make strong assumptions or don't scale to the sizes seen in public releases like Census data. By separating the imputation from the posterior sampling, this method lets users leverage standard Bayesian tools after a relatively simple first step. It builds directly on the Guha and Reiter idea but adds the asymptotic theory and the frequentist checks, which strengthens the case. The real-data example shows it can be used on actual products without needing entirely new machinery. The potential weak point is the reliance on those mild assumptions for the proof. The abstract does not spell out what they are in detail, so it is important to verify whether they cover the kinds of noise mechanisms and imputation procedures that arise in practice. If the assumptions are too restrictive for common DP implementations, the validity guarantee might not apply as broadly as hoped. The simulations are reported to support conservative behavior, but without more on the design, it is difficult to assess how robust that finding is across different settings. This work is aimed at statisticians and applied researchers who need to draw inferences from large-scale differentially private data products. Anyone dealing with policy-relevant data releases that use DP would find the method and the example relevant. It shows clear thinking about the problem and engages with the literature on private data analysis. I think it deserves a serious referee. The asymptotic result combined with the empirical checks and application provides enough substance for a full review. I would recommend sending this paper out for peer review.

Referee Report

1 major / 2 minor

Summary. The paper proposes a two-step approximate Bayesian method for inference with differentially privatized data: first impute the confidential data, then draw samples from the non-private posterior. The central claim is a proof that this approximate sampler is asymptotically valid under mild assumptions on the DP mechanism, the imputation procedure, and the data-generating process. Simulations are used to show that the method exhibits conservative frequentist coverage properties, and the approach is demonstrated on an analysis of homeownership drivers using the 2022 American Community Survey.

Significance. If the asymptotic validity result holds, the method supplies a scalable Bayesian approximation for large privatized datasets that avoids strong parametric assumptions and is motivated by but not limited to Bayesian theory. This is relevant for official statistics applications such as Census products, where DP noise complicates inference. The combination of a theoretical guarantee, simulation evidence of conservative frequentist behavior, and a real-data example constitutes a practical contribution to privacy-preserving statistical methodology.

major comments (1)

[Section 3] The proof of asymptotic validity (Section 3) is stated to hold under 'mild assumptions' on the differential privacy mechanism, imputation step, and underlying DGP. These assumptions need to be stated explicitly in a formal theorem, including any requirements on the privacy budget, the rate of imputation error, and regularity conditions on the posterior, so that readers can verify whether they are satisfied for typical privatized data products.

minor comments (2)

[Section 4] In the simulation study (Section 4), the description of how the imputation step is implemented and how the non-private posterior is sampled should include sufficient detail (e.g., software packages, number of imputations, MCMC settings) to support reproducibility.
[Section 5] The real-data application (Section 5) would benefit from a brief sensitivity analysis showing how results change under different privacy budgets or imputation models, to illustrate robustness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for the constructive recommendation of minor revision. We address the major comment below.

read point-by-point responses

Referee: [Section 3] The proof of asymptotic validity (Section 3) is stated to hold under 'mild assumptions' on the differential privacy mechanism, imputation step, and underlying DGP. These assumptions need to be stated explicitly in a formal theorem, including any requirements on the privacy budget, the rate of imputation error, and regularity conditions on the posterior, so that readers can verify whether they are satisfied for typical privatized data products.

Authors: We agree that explicitly stating the assumptions in a formal theorem will improve clarity and allow readers to assess applicability to specific privatized data products. In the revised version, Section 3 will be updated to include a dedicated theorem that formally lists all assumptions, covering requirements on the privacy budget (e.g., appropriate rates at which epsilon_n tends to zero), the imputation error rate (e.g., o_p(1) consistency), and standard regularity conditions on the posterior (such as consistency and asymptotic normality under the non-private model). This change strengthens the presentation without altering the underlying results. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central claim is an independent proof of asymptotic validity for the two-step approximate sampler (impute confidential data, then draw from the non-private posterior) under mild assumptions on the DP mechanism, imputation step, and DGP. This derivation does not reduce by the paper's own equations to a self-definitional equivalence, a fitted input renamed as prediction, or a self-citation chain. The method is explicitly inspired by external prior work (Guha and Reiter 2025) rather than justified solely by author-overlapping citations. Simulations are presented as supplementary frequentist checks rather than load-bearing for the validity result. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; specific free parameters, axioms, and invented entities cannot be enumerated without the full manuscript. The central claim rests on unspecified 'mild assumptions' for the asymptotic proof and on the validity of the imputation step.

pith-pipeline@v0.9.0 · 5487 in / 1155 out tokens · 40060 ms · 2026-05-08T02:11:57.170368+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 2 canonical work pages · 1 internal anchor

[1]

Abadi, M., A. Chu, I. Goodfellow, H. B. McMahan, I. Mironov, K. Talwar, and L. Zhang (2016). Deep learning with differential privacy. InProceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, pp. 308–318. Abowd, J., R. Ashmead, G. Simson, D. Kifer, P. Leclerc, A. Machanavajjhala, and W. Sex- ton (2019). Census TopDown: Diffe...

work page internal anchor Pith review Pith/arXiv arXiv 2016
[2]

Raskhodnikova, and A

Nissim, K., S. Raskhodnikova, and A. Smith (2007). Smooth sensitivity and sampling in private data analysis. InProceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, pp. 75–84. Parilla, J. and S. Liu (2018). Examining the local value of economic development incentives.Washington, DC: Brookings Institution. https://www. brookings. edu...

2007
[3]

Privacy Loss in Apple's Implementation of Differential Privacy on MacOS 10.12

Springer. Stone, L. C., B. Boursaw, S. P. Bettez, T. L. Marley, and H. Waitzkin (2015). Place as a predictor of health insurance coverage: A multivariate analysis of counties in the United States.Health & place 34, 207–214. Tang, J., A. Korolova, X. Bai, X. Wang, and X. Wang (2017). Privacy loss in Apple’s implementation of differential privacy on MacOS 1...

work page Pith review arXiv 2015

[1] [1]

Abadi, M., A. Chu, I. Goodfellow, H. B. McMahan, I. Mironov, K. Talwar, and L. Zhang (2016). Deep learning with differential privacy. InProceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, pp. 308–318. Abowd, J., R. Ashmead, G. Simson, D. Kifer, P. Leclerc, A. Machanavajjhala, and W. Sex- ton (2019). Census TopDown: Diffe...

work page internal anchor Pith review Pith/arXiv arXiv 2016

[2] [2]

Raskhodnikova, and A

Nissim, K., S. Raskhodnikova, and A. Smith (2007). Smooth sensitivity and sampling in private data analysis. InProceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, pp. 75–84. Parilla, J. and S. Liu (2018). Examining the local value of economic development incentives.Washington, DC: Brookings Institution. https://www. brookings. edu...

2007

[3] [3]

Privacy Loss in Apple's Implementation of Differential Privacy on MacOS 10.12

Springer. Stone, L. C., B. Boursaw, S. P. Bettez, T. L. Marley, and H. Waitzkin (2015). Place as a predictor of health insurance coverage: A multivariate analysis of counties in the United States.Health & place 34, 207–214. Tang, J., A. Korolova, X. Bai, X. Wang, and X. Wang (2017). Privacy loss in Apple’s implementation of differential privacy on MacOS 1...

work page Pith review arXiv 2015