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Differential privacy (DP) quantifies privacy loss by analyzing noise injected into output statistics. For non-trivial statistics, this noise is necessary to ensure finite privacy loss. However, data curators frequently release collections of statistics where some use DP mechanisms and others are released as-is, i.e., without additional randomized noise. Consequently, DP alone cannot characterize the privacy loss attributable to the entire collection of releases. In this paper, we present a privacy formalism, $(ε, \{ Θ_z\}_{z \in \mathcal{Z}})$-Pufferfish ($ε$-TP for short when $\{ Θ_z\}_{z \in \mathcal{Z}}$ is implied), a collection of Pufferfish mechanisms indexed by realizations of a random variable $Z$ representing public information not protected with DP noise. First, we prove that this definition has similar properties to DP. Next, we introduce mechanisms for releasing partially private data (PPD) satisfying $ε$-TP and prove their desirable properties. We provide algorithms for sampling from the posterior of a parameter given PPD. We then compare this inference approach to the alternative where noisy statistics are deterministically combined with Z. We derive mild conditions under which using our algorithms offers both theoretical and computational improvements over this more common approach. Finally, we demonstrate all the effects above on a case study on COVID-19 data.