Full Convergence of the Iterative Bayesian Update and Applications to Mechanisms for Privacy Protection

TL;DR

This paper corrects the theoretical understanding of the iterative Bayesian update (IBU), proves its convergence under general conditions, and demonstrates its effectiveness in privacy-preserving data recovery compared to matrix inversion methods.

Contribution

The paper provides a rigorous convergence proof for IBU under realistic data assumptions and extends its applicability to broader privacy models.

Findings

IBU outperforms INV on Geometric mechanisms

IBU is comparable to INV on $k$-RR and RAPPOR mechanisms

Theoretical flaws in previous IBU analyses are addressed

Abstract

The iterative Bayesian update (IBU) and the matrix inversion (INV) are the main methods to retrieve the original distribution from noisy data resulting from the application of privacy protection mechanisms. We show that the theoretical foundations of the IBU established in the literature are flawed, as they rely on an assumption that in general is not satisfied in typical real datasets. We then fix the theory of the IBU, by providing a general convergence result for the underlying Expectation-Maximization method. Our framework does not rely on the above assumption, and also covers a more general local privacy model. Finally we evaluate the precision of the IBU on data sanitized with the Geometric, $k$-RR, and RAPPOR mechanisms, and we show that it outperforms INV in the first case, while it is comparable to INV in the other two cases.