One Step to Efficient Synthetic Data

TL;DR

The paper investigates synthetic data generation and identifies a fundamental flaw in the common parametric bootstrap approach: sampling from a fitted model can yield inefficient estimators and a synthetic distribution that does not converge to the true data-generating process. It introduces a general one-step synthetic data method that preserves the efficiency of the original estimator and extends naturally to partially synthetic and differentially private fully synthetic data, with theoretical support via KL and TV convergence arguments. Empirical studies across Burr distributions, log-linear contingency tables, and DP beta and two-sample tests demonstrate that one-step synthetic data closely approximates the true distribution and improves inference relative to traditional parametric bootstrap, including in privacy-preserving settings. The work provides a practical, scalable alternative for synthetic data and hypothesis testing in models with intractable likelihoods, with broad applicability and clear DP integration paths for real-world data sharing and analysis.

Abstract

A common approach to synthetic data is to sample from a fitted model. We show that under general assumptions, this approach results in a sample with inefficient estimators and whose joint distribution is inconsistent with the true distribution. Motivated by this, we propose a general method of producing synthetic data, which is widely applicable for parametric models, has asymptotically efficient summary statistics, and is both easily implemented and highly computationally efficient. Our approach allows for the construction of both partially synthetic datasets, which preserve certain summary statistics, as well as fully synthetic data which satisfy the strong guarantee of differential privacy (DP), both with the same asymptotic guarantees. We also provide theoretical and empirical evidence that the distribution from our procedure converges to the true distribution. Besides our focus on synthetic data, our procedure can also be used to perform approximate hypothesis tests in the presence of intractable likelihood functions.

Figures (7)

• Figure 1: Empirical power of the Kolmogorov-Smirnov test for $\mathrm{Burr}(2,4)$ at type I error $.05$. $(X_i)$ are drawn i.i.d from $\mathrm{Burr}(2,4)$, $(Z_i)$ are drawn i.i.d from $\mathrm{Burr}\{{\hat{\theta}_X}\}$, and $(Y_i)$ are from Algorithm \ref{['alg:onestep']}. Results are averaged over 10000 replicates, for each $n$. Standard errors are approximately $0.0022$ for lines 1 and 3, and $0.0036$ for line 2.
• Figure 1: Average squared $\ell_2$-distance between the estimated parameters and the true parameters on the log-scale. Averages are over 200 replicates for both plots. ${\hat{\theta}_X}$ is from the true model, ${\hat{\theta}_Z}$ from the fitted model, and ${\hat{\theta}_Y}$ from Algorithm \ref{['alg:onestep']}.
• Figure 2: Recorded injuries according to seatbelt use, gender, and location. Source: agresti2003categorical. Originally credited to Cristanna Cook, Medical Care Development, Augusta, Maine.
• Figure 2: Simulations for the DP two sample proportion test of Section \ref{['s:DPtesting']}. In red is the one-step test, and in blue is the parametric bootstrap test (abbreviated as PB). Sample sizes are $n=m=200$, privacy parameter is $\epsilon=1$, and type I error is $.05$.
• Figure 3: Additional simulations for Section 6.3. In normal reading order, $\epsilon=.5,2,4,\infty$. Note that Figure 1(b) in the main paper is for $\epsilon=1$

Theorems & Definitions (17)

• Example 1
• Definition 1: Local Asymptotic Equivariance
• Theorem 1
• Example 2
• Theorem 2
• Lemma 1
• Theorem 3
• Definition 2: Differential privacy: dwork2006calibrating
• Proposition 1: Sensitivity and Laplace Mechanism: dwork2006calibrating
• proof : Proof of Theorem 1