# bounding min-entropy gain in differential privacy

In privacy-related computer science literature, we say that a randomized algorithm $$\mathcal{K}$$ that produces a model $$\theta$$ from a sample $$X=(x_1,...,x_n)$$ is $$\epsilon$$-differentially private iff

$$\forall \theta, \quad \mathbb{P}[\mathcal{K}(X)=\theta | X=X_0] \leq e^\epsilon \mathbb{P}[\mathcal{K} (X)=\theta|X=X_1] \tag{1}$$

where $$X_0$$ and $$X_1$$ are any Hamming-1 neighbors (differ by a single entry)

Using Bayes rule, we can rewrite (1) as a bounded Bayes ratio

$$\frac{\mathbb{P}[X=X_0|\theta]}{\mathbb{P}[X=X_1|\theta]} \leq e^\epsilon \frac{\mathbb{P}[X=X_0]}{\mathbb{P}[X=X_1]} \tag{2}$$

In order to relate this to the Bayes (irreducible) error of an adversary trying to estimate $$X$$ from $$\theta$$, one can leverage the min-entropy

$$H_\infty(X) = -\log \max_x \mathbb{P}[X=x] \\ \text{and} \\ H_\infty(X|\theta) = -\log \mathbb{E}_\theta \max_x \mathbb{P}[X=x|\theta]$$

First part of the question : Can we translate (2) into a bound involving $$H_\infty(X|\theta)$$ and $$H_\infty(X)$$ ?

Second part of the question : Since $$H_\infty(X|\theta)$$ takes the expectation over $$\theta$$, it looks like a weaker guarantee than (1), (bounded leakage only holding in expectation, with some $$\theta$$ leaking more than the bound). Is there a well-accepted entropy definition that keeps the same expressivity as (1), e.g.,

$$H_\text{bla}(X|\theta) = -\log \max_{\theta,x} \mathbb{P}[X=x|\theta]$$

or something like that ?

thanks !