  Privacy protection methods, such as differentially private mechanisms,
introduce noise into resulting statistics which often produces complex and
intractable sampling distributions. In this paper, we propose a
simulation-based "repro sample" approach to produce statistically valid
confidence intervals and hypothesis tests, which builds on the work of Xie and
Wang (2022). We show that this methodology is applicable to a wide variety of
private inference problems, appropriately accounts for biases introduced by
privacy mechanisms (such as by clamping), and improves over other
state-of-the-art inference methods such as the parametric bootstrap in terms of
the coverage and type I error of the private inference. We also develop
significant improvements and extensions for the repro sample methodology for
general models (not necessarily related to privacy), including 1) modifying the
procedure to ensure guaranteed coverage and type I errors, even accounting for
Monte Carlo error, and 2) proposing efficient numerical algorithms to implement
the confidence intervals and $p$-values.
