pufferfish privacy
This post described the concept of the Pufferfish framework introduced here^{1}.
We look over notations first:
 \(\mathcal{I}\): The set of possible database instances.
 \(\mathfrak{Data}\): A random variable for the true dataset, which the attacker has no knowledge of.
 \(\mathbb{S}\): A set of potential secrets, revealing which may cause harm.
 \(\mathbb{D}\): A conservative collection of plausible data generating distributions, i.e the set of evolution scenarios.
 \(\theta\): A probability distribution.
The Pufferfish framework needs specification of three components:
 \(\mathbb{S}\): This is the explicit specification of what we would want to protect. Any statement \(s \in \mathbb S\) may or may not be true. Essentially, \(s\) should be in \(\mathbb S\), if the release of the statement \(s\) or \(\neg s\) is harmful.
 \(\mathbb{S}_{pairs} \subseteq \mathbb{S} \times \mathbb{S}\): This tells us how to protect \(\mathbb{S}\). Given a discriminative pair \((s_i, s_j)\), the attacker mustn’t be able to know if \(s_i\) is the true data, or if \(s_j\) is the true data. Thus, \(s_i\) and \(s_j\) must be mutually exclusive but can be nonexhaustive.
 \(\mathbb{D}\): This is a set of conservative assumptions about how data is evolved/generated, and knowledge about potential attackers. Thus, \(\mathbb{D}\) is a set of probability distributions over \(\mathcal{I}\) and each \(\theta \in \mathbb D\) corresponds to an attacker we want to protect against, and is the belief of the attacker on how the data is generated.
Definition 1: Given a set of potential secrets \(\mathbb{S}\), a set of discriminative pairs \(\mathbb{S}_{pairs}\), a set of data evolution scenarios \(\mathbb{D}\), and a privacy parameter \(\epsilon > 0\), an algorithm \(\mathfrak{M}\) satsifies \(\epsilon\)PufferFish\((\mathbb{S}, \mathbb{S}_{pairs}, \mathbb{D})\) privacy if
\(\triangleright \;\forall \; \omega \in\) range\((\mathfrak{M})\)
\(\triangleright \;\forall \;\) pairs \((s_i, s_j) \in \mathbb{S}_{pairs}\)
\(\triangleright \;\forall \;\) distributions \(\theta \in \mathbb D\) for which Pr\((s_i\theta)\neq 0\) and Pr\((s_j\theta) \neq 0\)
the following holds

Kifer, Daniel, and Ashwin Machanavajjhala. “Pufferfish: A framework for mathematical privacy definitions.” ACM Transactions on Database Systems (TODS) 39.1 (2014): 136. ↩