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I know that we need some sort of sensitivity(global, local) to calculate noise that needs to be added for differential privacy. The noise is the maximum difference between two neighboring datasets. However, if I am working in probability spaces, i.e. my values are bounded by 0,1. My sensitivity would be 1, which seems too large for adding noise to the probabilities.

I tried looking up any relevant literature that provides any tighter upper bounds, but no success. I am only interested in adding noise to the probabilities, not to the mechanism that they are generated from.

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Do you need local differential privacy (i.e. no central server knows the real data of individuals) or global differential privacy (where you have all the real data, and want to generate differentially private statistics out of it)?

In the former case, you need to add a significant amount of noise (typically Laplace noise) to each probability, and there's basically no better way of doing it. This will mean that every data point will be very noisy, but if you have enough of them, you might be able to retrieve some information. The "amount of noise" will be similar to the randomized response mechanism described here or here.

In the latter case, it depends on what you're computing. If you're computing the average probability, the noise you add can be scaled by $1/n$, where $n$ is the total number of users. See for example the methods presented here, in the "summing or averaging numbers" section.

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