Differential Confounding Privacy and Inverse Composition

Differential privacy (DP) has become the gold standard for privacy-preserving data analysis, but its applicability can be limited in scenarios involving complex dependencies between sensitive information and datasets. To address this, we introduce \textit{differential confounding privacy} (DCP), a specialized form of the Pufferfish privacy (PP) framework that generalizes DP by accounting for broader relationships between sensitive information and datasets. DCP adopts the $(\epsilon, \delta)$-indistinguishability framework to quantify privacy loss. We show that while DCP mechanisms retain privacy guarantees under composition, they lack the graceful compositional properties of DP. To overcome this, we propose an \textit{Inverse Composition (IC)} framework, where a leader-follower model optimally designs a privacy strategy to achieve target guarantees without relying on worst-case privacy proofs, such as sensitivity calculation. Experimental results validate IC's effectiveness in managing privacy budgets and ensuring rigorous privacy guarantees under composition.