Fisher information under local differential privacy

We develop data processing inequalities that describe how Fisher information from statistical samples can scale with the privacy parameter <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula> under local differential privacy constraints. These bounds are valid under general conditions on the distribution of the score of the statistical model, and they elucidate under which conditions the dependence on <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula> is linear, quadratic, or exponential. We show how these inequalities imply order-optimal lower bounds for private estimation for both the Gaussian location model and discrete distribution estimation for all levels of privacy <inline-formula> <tex-math notation="LaTeX">$\varepsilon >0$ </tex-math></inline-formula>. We further apply these inequalities to sparse Bernoulli models and demonstrate privacy mechanisms and estimators with order-matching squared <inline-formula> <tex-math notation="LaTeX">$\ell ^{2}$ </tex-math></inline-formula> error.