Joint Distribution Analysis of Multi-Dimensional Randomized Response
Wenli Wang, Yijun Zhang, Jun Yan, Yihui Zhou, Laifeng Lu · 2023
Data collection can provide services and convenience for people, but disclosure of personal privacy during data collection can be harmful. So, it is needed to protect personal privacy while collecting data. Randomized response is a reliable privacy-preserving technique. For multi-dimensional data, the most direct mechanism is RR-Independent, which applies randomized response on each attribute independently. After applying the RR-independent mechanism, the joint distribution of perturbed data needs to be modified in order to be close to that of input data better. Two estimation methods of joint distribution, Naive estimation and Castell estimation, are studied in this paper. Then, we propose a sufficient and necessary condition for the two estimations to be equal. In addition, a mathematical study shows that Castell estimation is closer to the original distribution of data than that of Naive estimation if the condition is not satisfied. Finally, numerical experiments are carried out to show that Castell estimation is better than Naive estimation and the results obtained by the two protocols in the paper are consistent.