On the `Semantics' of Differential Privacy: A Bayesian Formulation (0803.3946v4)

Abstract: Differential privacy is a definition of "privacy'" for algorithms that analyze and publish information about statistical databases. It is often claimed that differential privacy provides guarantees against adversaries with arbitrary side information. In this paper, we provide a precise formulation of these guarantees in terms of the inferences drawn by a Bayesian adversary. We show that this formulation is satisfied by both "vanilla" differential privacy as well as a relaxation known as (epsilon,delta)-differential privacy. Our formulation follows the ideas originally due to Dwork and McSherry [Dwork 2006]. This paper is, to our knowledge, the first place such a formulation appears explicitly. The analysis of the relaxed definition is new to this paper, and provides some concrete guidance for setting parameters when using (epsilon,delta)-differential privacy.

• The paper introduces a Bayesian formulation of differential privacy that quantifies privacy by comparing posterior distributions with and without individual data.
• It analyzes both strict and relaxed DP frameworks, showing how precise parameter calibration can maintain privacy against adversaries with side information.
• The study presents a key Conditioning Lemma that strengthens semantic privacy guarantees and offers practical insights for designing robust privacy-preserving algorithms.

A Bayesian Formulation of Differential Privacy: Analysis and Implications

This paper presents a Bayesian interpretation of differential privacy (DP), providing precise formulations of privacy guarantees against adversaries equipped with arbitrary side information. Traditional definitions of differential privacy often leave the underlying assumptions and the practical meaning of "privacy" somewhat ambiguous. By focusing on a Bayesian adversary model, the paper elucidates these guarantees through a more formalized approach, assessing how DP withstands inference attacks.

Core Contributions

1. Bayesian Formulation of Differential Privacy: The paper introduces a Bayesian formulation of DP, which aligns with the natural understanding of privacy as an adversary being unable to draw significantly different conclusions with and without an individual's data. This is quantified by examining the statistical difference between posterior distributions of outcomes based on an adversary's prior beliefs and sanitized outputs generated by DP mechanisms.
2. Analysis of Relaxed Differential Privacy Notions: The traditional $$ allows for a small probability of breach in privacy guarantees. The authors provide a comprehensive examination of how these relaxed definitions can maintain privacy under Bayesian scrutiny, as long as the parameters are meticulously calibrated.
3. Semantic Interpretations and Implications: By bridging the gap between semantic privacy definitions and differential privacy, the paper asserts that DP provides guarantees that are valid even under substantial adversarial side information. This theoretical contribution has practical implications for designing databases and algorithms where privacy assurance needs to be clear and justified.
4. Mathematical Results and Conditioning Lemma: The paper introduces a Conditioning Lemma, a pivotal tool in proving that differentially private algorithms adhere to the semantic privacy guarantees. This lemma is significant for understanding the behavior of Bayesian adversaries and their belief updating processes post observation of outputs generated under differential privacy.

Theoretical and Practical Implications

Theoretical Impact:

The Bayesian interpretation strengthens the theoretical foundations of differential privacy. It solves ambiguities about the meaning of privacy when auxiliary information is present. The paper bridges conceptual gaps, offering a refined way to understand privacy through statistical distances and Bayesian inference.

Practical Considerations:

From a practical standpoint, this paper's insights are invaluable for practitioners designing privacy-preserving algorithms. The results guide the setting of parameters in $(,\delta)$-DP to ensure robustness against inference attacks, balancing between data utility and privacy assurance. The work underscores the need for carefully choosing $\delta$, especially in applications where datasets have sensitive information.

Future Directions and Speculation

The Bayesian analysis opens avenues for future research in privacy preservation. One potential direction is the exploration of differential privacy mechanisms under more complex adversarial models, further integrating machine learning and statistical decision theory. Another pivotal area is refining $(,\delta)$-DP relaxations to optimize their utility-privacy tradeoffs while maintaining rigorous guarantees against Bayesian adversaries.

Moreover, exploring combinations with cryptographic techniques could yield hybrid models enhancing both theoretical guarantees and practical applicability, especially in distributed and federated learning contexts.

In conclusion, this paper represents a rigorous step toward formalizing and understanding differential privacy within a Bayesian framework. It provides the necessary theoretical tools and insights for improved privacy mechanisms, both now and as the landscape of data privacy challenges continues to evolve.