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.

翻译：我们研究了对封闭数据进行有区别的私下分析的两种方法,并将这些方法扩大到非否定性查询。我们首先回顾,对于 Laplace 机制而言,对非否定性查询和缩短适用的边界过宽(BIT)都会导致严格的积极偏差。然后我们考虑使用翻译的斜坡功能对双边投资条约进行概括化。我们明确将这一类中的最佳功能定性为最坏的偏差。我们表明,将任何可平方的后处理功能应用到 Laplace 机制会导致严格的积极最大绝对偏差。同样,我们称之为限制的脱节一般化也显示了相应的结果。我们还简要地考虑了基于积极数据的多重复制机制的替代方法,并表明,如果没有额外的限制,这些机制可能导致无限的偏差。