  We explore and compare a variety of definitions for privacy and disclosure
limitation in statistical estimation and data analysis, including (approximate)
differential privacy, testing-based definitions of privacy, and posterior
guarantees on disclosure risk. We give equivalence results between the
definitions, shedding light on the relationships between different formalisms
for privacy. We also take an inferential perspective, where---building off of
these definitions---we provide minimax risk bounds for several estimation
problems, including mean estimation, estimation of the support of a
distribution, and nonparametric density estimation. These bounds highlight the
statistical consequences of different definitions of privacy and provide a
second lens for evaluating the advantages and disadvantages of different
techniques for disclosure limitation.
