  The simplest and most widely applied method for guaranteeing differential
privacy is to add instance-independent noise to a statistic of interest that is
scaled to its global sensitivity. However, global sensitivity is a worst-case
notion that is often too conservative for realized dataset instances. We
provide methods for scaling noise in an instance-dependent way and demonstrate
that they provide greater accuracy under average-case distributional
assumptions.
  Specifically, we consider the basic problem of privately estimating the mean
of a real distribution from i.i.d.~samples. The standard empirical mean
estimator can have arbitrarily-high global sensitivity. We propose the trimmed
mean estimator, which interpolates between the mean and the median, as a way of
attaining much lower sensitivity on average while losing very little in terms
of statistical accuracy.
  To privately estimate the trimmed mean, we revisit the smooth sensitivity
framework of Nissim, Raskhodnikova, and Smith (STOC 2007), which provides a
framework for using instance-dependent sensitivity. We propose three new
additive noise distributions which provide concentrated differential privacy
when scaled to smooth sensitivity. We provide theoretical and experimental
evidence showing that our noise distributions compare favorably to others in
the literature, in particular, when applied to the mean estimation problem.
