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Obtaining $(ε,δ)$-differential privacy guarantees when using a Poisson mechanism to synthesize contingency tables

James Jackson, Robin Mitra, Brian Francis, Iain Dove

TL;DR

This work investigates obtaining formal privacy guarantees when perturbing contingency-table counts with a Poisson synthesis mechanism. It shows that pure $\epsilon$-DP cannot be achieved with Poisson, but the $(\epsilon,\delta)$-probabilistic DP relaxation can be obtained by leveraging the Poisson CDF, with explicit bounds that depend on the tuning parameter $\alpha$ and the original counts. The authors provide analytical expressions for $1-\delta$ (e.g., $1-\delta = F_{1+\alpha}^P\left[\frac{1+\epsilon}{\log\left(\frac{1+\alpha}{\alpha}\right)}\right]$ for $\epsilon>1$) and illustrate the approach on an English School Census–like administrative dataset, highlighting the privacy–utility trade-offs. The work suggests that richer count distributions (e.g., negative binomial) may yield better utility while preserving DP-type guarantees and explains why multinomial–Dirichlet mechanisms can attain $\epsilon$-DP in some settings, in contrast to Poisson-based methods.

Abstract

We show that differential privacy type guarantees can be obtained when using a Poisson synthesis mechanism to protect counts in contingency tables. Specifically, we show how to obtain $(ε, δ)$-probabilistic differential privacy guarantees via the Poisson distribution's cumulative distribution function. We demonstrate this empirically with the synthesis of an administrative-type confidential database.

Obtaining $(ε,δ)$-differential privacy guarantees when using a Poisson mechanism to synthesize contingency tables

TL;DR

This work investigates obtaining formal privacy guarantees when perturbing contingency-table counts with a Poisson synthesis mechanism. It shows that pure -DP cannot be achieved with Poisson, but the -probabilistic DP relaxation can be obtained by leveraging the Poisson CDF, with explicit bounds that depend on the tuning parameter and the original counts. The authors provide analytical expressions for (e.g., for ) and illustrate the approach on an English School Census–like administrative dataset, highlighting the privacy–utility trade-offs. The work suggests that richer count distributions (e.g., negative binomial) may yield better utility while preserving DP-type guarantees and explains why multinomial–Dirichlet mechanisms can attain -DP in some settings, in contrast to Poisson-based methods.

Abstract

We show that differential privacy type guarantees can be obtained when using a Poisson synthesis mechanism to protect counts in contingency tables. Specifically, we show how to obtain -probabilistic differential privacy guarantees via the Poisson distribution's cumulative distribution function. We demonstrate this empirically with the synthesis of an administrative-type confidential database.
Paper Structure (8 sections, 1 theorem, 16 equations, 3 figures)

This paper contains 8 sections, 1 theorem, 16 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Terminology and definitions
  3. Examples of existing DP mechanisms
  4. Satisfying $(\epsilon, \delta)$-probabilistic DP with a Poisson synthesis mechanism
  5. An empirical example
  6. The English School Census administrative database
  7. Applying the Poisson synthesis mechanism
  8. Discussion

Key Result

Theorem 1

If a perturbation mechanism $\mathcal{M}$ satisfies $(\epsilon, \delta)$-probabilistic DP, then it also satisfies $(\epsilon, \delta)$-DP. (Proof: see Goetz2012)

Figures (3)

  • Figure 1: The relationship between $\alpha$ and $\delta$ in the Poisson synthesis mechanism for $\epsilon=1.5$ and $\epsilon=3$.
  • Figure 2: Combinations of $\delta$ such that $(\epsilon, \delta)$-probabilistic DP is achieved when the Poisson is used, for various max$_i a_i$ and $\epsilon$ equal to 1.5, 2, 2.5 and 3.
  • Figure 3: For different values of $\alpha$, boxplots showing percentage differences between original and synthetic counts (utility) for original counts in the range 1--10.

Theorems & Definitions (7)

  • Definition 1: $\epsilon$-DP
  • Definition 2: $(\epsilon, \delta)$-DP
  • Definition 3: $(\epsilon, \delta)$-probabilistic DP
  • Theorem 1: $(\epsilon, \delta)$-probabilistic DP implies $(\epsilon, \delta)$-DP
  • Example 1: The Laplace mechanism
  • Example 2: The Gaussian mechanism
  • Example 3: Multinomial-Dirichlet synthesizer