  One goal of statistical privacy research is to construct a data release
mechanism that protects individual privacy while preserving information
content. An example is a {\em random mechanism} that takes an input database
$X$ and outputs a random database $Z$ according to a distribution
$Q_n(\cdot|X)$. {\em Differential privacy} is a particular privacy requirement
developed by computer scientists in which $Q_n(\cdot |X)$ is required to be
insensitive to changes in one data point in $X$. This makes it difficult to
infer from $Z$ whether a given individual is in the original database $X$. We
consider differential privacy from a statistical perspective. We consider
several data release mechanisms that satisfy the differential privacy
requirement. We show that it is useful to compare these schemes by computing
the rate of convergence of distributions and densities constructed from the
released data. We study a general privacy method, called the exponential
mechanism, introduced by McSherry and Talwar (2007). We show that the accuracy
of this method is intimately linked to the rate at which the probability that
the empirical distribution concentrates in a small ball around the true
distribution.
