  The iterative Bayesian update (IBU) and the matrix inversion (INV) are the
main methods to retrieve the original distribution from noisy data resulting
from the application of privacy protection mechanisms. We show that the
theoretical foundations of the IBU established in the literature are flawed, as
they rely on an assumption that in general is not satisfied in typical real
datasets. We then fix the theory of the IBU, by providing a general convergence
result for the underlying Expectation-Maximization method. Our framework does
not rely on the above assumption, and also covers a more general local privacy
model. Finally we evaluate the precision of the IBU on data sanitized with the
Geometric, $k$-RR, and RAPPOR mechanisms, and we show that it outperforms INV
in the first case, while it is comparable to INV in the other two cases.
