epsilon (ε) privacy
**Epsilon (ε) privacy** is the core parameter of **differential privacy** — it quantifies the **maximum privacy loss** that any individual can experience from their data being included in a computation. A smaller epsilon means **stronger privacy protection** but typically comes at the cost of reduced data utility.
**Formal Definition**
A mechanism M satisfies ε-differential privacy if for any two neighboring datasets D and D' (differing in one person's data) and any possible output S:
$$P[M(D) \in S] \leq e^\varepsilon \cdot P[M(D') \in S]$$
This means the **output distribution changes by at most a factor of $e^\varepsilon$** whether or not any individual's data is included.
**Interpreting Epsilon**
- **ε = 0**: Perfect privacy — the output reveals absolutely nothing about any individual. But provides no utility.
- **ε = 0.1**: Very strong privacy — an attacker gains at most ~10% more information from the output.
- **ε = 1**: Moderate privacy — standard benchmark for "good" differential privacy.
- **ε = 10**: Weak privacy protection — often considered the upper bound for meaningful privacy.
- **ε → ∞**: No privacy — output directly reveals the data.
**Privacy Budget**
- Each query or computation on the data "spends" some epsilon from the privacy budget.
- **Composition Theorem**: Running k analyses on the same data costs approximately ε × √k total privacy (under advanced composition).
- Once the budget is exhausted, no more queries should be answered to maintain privacy guarantees.
**Practical Usage**
- **Apple**: Uses ε = 2–8 for collecting emoji and typing statistics in iOS.
- **Google**: Uses ε = 2–9 for Chrome usage statistics via **RAPPOR**.
- **US Census**: Applied differential privacy with aggregated ε budgets for the 2020 Census.
**The Privacy-Utility Trade-Off**
Smaller ε requires adding **more noise**, which reduces the accuracy of results. Choosing ε involves balancing privacy protection against the need for useful, accurate outputs — a fundamental design decision with no universally correct answer.