# The amount of noise needed for independent vs. dependent features in differential privacy

There is this paper that says:

Remark 1. DP noise is applied with the assumption that features are independent from each other, meaning a maximal amount of noise must be applied to each feature to ensure DP. With knowledge of feature dependence, hypothetically less noise can be applied to the dependent features as there is less uniquely identifying information between the dependent features.

Is this true? or the opposite?

For example, let 2 binary dependent features X1, X2 having the relation X2=1-X1

Then using a DP method (like randomized response) one releases X1=1, X2=1

No matter what is the amount of noise, the adversary is sure that either X1 or X2 must show the true value because of dependency. If X1 and X2 are not dependent, this wasn't the case.

So, doesn't dependency get it worse?

One may say in this case we can only publish one of X1 or X2 because one of them can be calculated based on another one. But in real world we mostly just know that features are dependent and don't know how exactly and to what extent they are dependant.

Thus, in general is that Remark 1 above true?