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Consider only $\epsilon$-differential privacy. The textbook definition for this is:

Definition 1: "A randomized algorithm $\mathcal{M}$ with domain $\mathbb{N}^{|\chi|}$ is $\epsilon$-differentially private if for all $S \subseteq Range(\mathcal{M})$ and for all $x, y \in \mathcal{N}^{|\chi|}$ such that $||x − y||_1 ≤ 1$:

$Pr[\mathcal{M}(x) \in S] ≤ \exp(\epsilon) Pr[\mathcal{M}(y) \in S]$."

However, in some papers, the definition is (incorrectly) written as:

Definition 2: "A randomized...is $\epsilon$-differentially private...if for all $S \in Range(\mathcal{M})$...:

$Pr[\mathcal{M}(x) = S] \leq \exp(\epsilon) Pr[\mathcal{M}(y) = S]$."

Obviously, the second definition is meaningless given continuous probability distributions. But suppose for now that we are working with discrete distributions so that the second definition has some meaning.

Question: In the case when we are working with discrete probability distributions, is there a problem with Definition 2? If so, what is the problem with it?

I seem to remember that there is still an issue with Definition 2 in the discrete case, but I've forgotten what the issue is. Please comment if any part of this question is unclear.

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If you are working with discrete probability distributions, then for any subset $S$ of the range, you have $$ \Pr[\mathcal{M}(x)\in S] = \sum_{s\in S} \Pr[\mathcal{M}(x)=s] \leq \sum_{s\in S} e^\varepsilon \Pr[\mathcal{M}(x')=s] = e^\varepsilon \Pr[\mathcal{M}(x')\in S] $$ and so the second definition implies the first (and thus they are equivalent).

You will have some issues with $(\varepsilon,\delta)$-differential privacy though, as the $\delta$'s will add up.


Technically, I guess that for both definitions you would need to write "measurable" wherever needed, as not all subsets/singletons may be measurable.

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