  The goal of this paper is to develop a practical and general-purpose approach
to construct confidence intervals for differentially private parametric
estimation. We find that the parametric bootstrap is a simple and effective
solution. It cleanly reasons about variability of both the data sample and the
randomized privacy mechanism and applies "out of the box" to a wide class of
private estimation routines. It can also help correct bias caused by clipping
data to limit sensitivity. We prove that the parametric bootstrap gives
consistent confidence intervals in two broadly relevant settings, including a
novel adaptation to linear regression that avoids accessing the covariate data
multiple times. We demonstrate its effectiveness for a variety of estimators,
and find that it provides confidence intervals with good coverage even at
modest sample sizes and performs better than alternative approaches.
