  We describe Bayesian inference for the mean and variance of bounded data
protected by differential privacy and modeled as Gaussian. Using this setting,
we demonstrate that analysts can and should take the constraints imposed by the
bounds into account when specifying prior distributions. Additionally, we
provide theoretical and empirical results regarding what classes of default
priors produce valid inference for a differentially private release in settings
where substantial prior information is not available. We discuss how these
results can be applied to Bayesian inference for regression with differentially
private data.
