  We study two methods for differentially private analysis of bounded data and
extend these to nonnegative queries. We first recall that for the Laplace
mechanism, boundary inflated truncation (BIT) applied to nonnegative queries
and truncation both lead to strictly positive bias. We then consider a
generalization of BIT using translated ramp functions. We explicitly
characterise the optimal function in this class for worst case bias. We show
that applying any square-integrable post-processing function to a Laplace
mechanism leads to a strictly positive maximal absolute bias. A corresponding
result is also shown for a generalisation of truncation, which we refer to as
restriction. We also briefly consider an alternative approach based on
multiplicative mechanisms for positive data and show that, without additional
restrictions, these mechanisms can lead to infinite bias.
