  We initiate an investigation of private sampling from distributions. Given a
dataset with $n$ independent observations from an unknown distribution $P$, a
sampling algorithm must output a single observation from a distribution that is
close in total variation distance to $P$ while satisfying differential privacy.
Sampling abstracts the goal of generating small amounts of realistic-looking
data. We provide tight upper and lower bounds for the dataset size needed for
this task for three natural families of distributions: arbitrary distributions
on $\{1,\ldots ,k\}$, arbitrary product distributions on $\{0,1\}^d$, and
product distributions on $\{0,1\}^d$ with bias in each coordinate bounded away
from 0 and 1. We demonstrate that, in some parameter regimes, private sampling
requires asymptotically fewer observations than learning a description of $P$
nonprivately; in other regimes, however, private sampling proves to be as
difficult as private learning. Notably, for some classes of distributions, the
overhead in the number of observations needed for private learning compared to
non-private learning is completely captured by the number of observations
needed for private sampling.
