- Thesis:
- Master in Mathematics
- Author:
- Dennis Reichard
- Title:
- Non-parametric Density Estimation under Local α-differential Privacy Constraints using Wavelet Techniques
- Supervisor:
- Jan JOHANNES
- Abstract:
- In our modern society, in which personal data has become a valuable good, two different interests have got into conflict with each other: On the one hand there is the interest of gathering such data and analyse it to gain scientific insights like the probability densities that determine the distribution of certain quantities. On the other hand there is the interest of the individuals to maintain control over their own data. Cristina Butucea, Amandine Dubois, Martin Kroll, and Adrien Saumard tried with their paper “Local Differential Privacy: Elbow Effect in Optimal Density Estimation and Adaptation over Besov Ellipsoids” to align these two interests a little more. This work will reproduce large parts of that paper in a more detailed form. In the first part some lower bounds for the minimax rates of convergence of all estimators satisfying certain quantified privacy constraints are developed. Furthermore two such privacy mechanisms are proposed as well as two density estimators based on them and their performance is studied and compared with the bounds shown before. These privacy mechanisms are based on the idea of estimating the wavelet coefficients with respect to a previously fixed wavelet basis based on every single data point individually, and to anonymise them by adding some appropriately scaled Laplace noise to these estimations before releasing them. This allows inference of the concrete data points based on the released coefficients only up to some previously fixed degree of certainty. But if one takes many estimations of the same coefficient based on the different data points all disturbed in this way together, one can gain an estimation for that coefficient where the impact of the added noise due to the law of large numbers decreases more and more with an increasing sample size. In the last part, which is not directly based on the mentioned paper, the results of a simulation study to these privacy mechanisms and density estimators are presented, their suitability for daily use is discussed and some possible modifications of them are proposed.
- Reference:
- C. Butucea, A. Dubois, M. Kroll, and A. Saumard. Local differential privacy: Elbox effect in optimal density estimation adaptation over Besov ellipsoids. Bernoulli, 26(3):1727-1764, 2020.
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |