  We investigate the local differential privacy (LDP) guarantees of a
randomized privacy mechanism via its contraction properties. We first show that
LDP constraints can be equivalently cast in terms of the contraction
coefficient of the $E_\gamma$-divergence. We then use this equivalent formula
to express LDP guarantees of privacy mechanisms in terms of contraction
coefficients of arbitrary $f$-divergences. When combined with standard
estimation-theoretic tools (such as Le Cam's and Fano's converse methods), this
result allows us to study the trade-off between privacy and utility in several
testing and minimax and Bayesian estimation problems.
