  The local privacy mechanisms, such as k-RR, RAPPOR, and the
geo-indistinguishability ones, have become quite popular thanks to the fact
that the obfuscation can be effectuated at the users end, thus avoiding the
need of a trusted third party. Another important advantage is that each data
point is sanitized independently from the others, and therefore different users
may use different levels of obfuscation depending on their privacy
requirements, or they may even use entirely different mechanisms depending on
the services they are trading their data for. A challenging requirement in this
setting is to construct the original distribution on the users sensitive data
from their noisy versions. Existing techniques can only estimate that
distribution separately on each obfuscation schema and corresponding noisy data
subset. But the smaller are the subsets, the more imprecise the estimations
are. In this paper we study how to avoid the subsets-fractioning problem when
combining local privacy mechanisms, thus recovering an optimal utility. We
focus on the estimation of the original distribution, and on the two main
methods to estimate it: the matrix-inversion method and the iterative Bayes
update. We consider various cases of combination of local privacy mechanisms,
and compare the flexibility and the performance of the two methods.
