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The local privacy mechanisms, such as k-RR, RAPPOR, and thegeo-indistinguishability ones, have become quite popular thanks to the factthat the obfuscation can be effectuated at the users end, thus avoiding theneed of a trusted third party. Another important advantage is that each datapoint is sanitized independently from the others, and therefore different usersmay use different levels of obfuscation depending on their privacyrequirements, or they may even use entirely different mechanisms depending onthe services they are trading their data for. A challenging requirement in thissetting is to construct the original distribution on the users sensitive datafrom their noisy versions. Existing techniques can only estimate thatdistribution separately on each obfuscation schema and corresponding noisy datasubset. But the smaller are the subsets, the more imprecise the estimationsare. In this paper we study how to avoid the subsets-fractioning problem whencombining local privacy mechanisms, thus recovering an optimal utility. Wefocus on the estimation of the original distribution, and on the two mainmethods to estimate it: the matrix-inversion method and the iterative Bayesupdate. We consider various cases of combination of local privacy mechanisms,and compare the flexibility and the performance of the two methods.

Reconstruction of the distribution of sensitive data under free-will privacy