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I am currently working with differential privacy and have a question regarding the composition of two DP mechanisms. In particular I am considering the following problem: consider having obtained a result $S$ of a DP mechanism $M_1(D)$ which satisfies $(\epsilon_1, \delta_1)$ differential privacy (for example via a Gaussian Mechanism). Now lets say I want to perform some mechanism $M_2$ on $S$, i.e. $M_2(S)$. Is there some bound I can give on what $M_2$ does with the privacy parameters of $M_1$? In other words can I amplify the privacy of a already private mechanism post-hoc?

Essentially the question boils down to the post-processing property of DP. Specifically, we know that we cannot worsen the privacy guarantee in post, but is there a way to amplify it after DP protection has already been applied?

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