Perfect Privacy and Strong Stationary Times for Markovian Sources
Fangwei Ye, Zonghong Liu, Parimal Parag, Salim El Rouayheb
TL;DR
This work addresses sharing data generated by a Markov chain while perfectly protecting the initial state $X_0$. It introduces two erasure-based mechanisms that implement data-dependent window redactions: an SST-based mechanism for transitively invariant chains and a sequential SMR mechanism for general chains, both yielding perfect privacy with a distortion that does not grow with the data length $N$. The SST-based scheme provides a closed-form distortion and relies on an optimal strong stationary time, while the SMR mechanism offers a window interpretation and a spectral bound showing a constant-average number of redacted samples dependent only on the transition matrix. Collectively, the results establish information-theoretic selective privacy guarantees for correlated data sharing and connect privacy to classical SST theory, with practical implications for privacy-preserving data sharing in regulated environments.
Abstract
We consider the problem of sharing correlated data under a perfect information-theoretic privacy constraint. We focus on redaction (erasure) mechanisms, in which data are either withheld or released unchanged, and measure utility by the average cardinality of the released set, equivalently, the expected Hamming distortion. Assuming the data are generated by a finite time-homogeneous Markov chain, we study the protection of the initial state while maximizing the amount of shared data. We establish a connection between perfect privacy and window-based redaction schemes, showing that erasing data up to a strong stationary time preserves privacy under suitable conditions. We further study an optimal sequential redaction mechanism and prove that it admits an equivalent window interpretation. Interestingly, we show that both mechanisms achieve the optimal distortion while redacting only a constant average number of data points, independent of the data length~$N$.