# Laplace mechanism from Exponential mechanism in Differential Privacy

In McSherry and Talwar's paper which introduces the exponential mechanism for differential privacy, they say that the Laplace mechanism can be captured by choosing the score function as $$q(d,r) = - |f(d) - r|$$ in the exponential mechanism.

However, choosing such a score function creates the following mechanism (assuming a uniform base measure $$\mu$$):

$$\mathcal{E}^\epsilon_q(d) := \text{choose r with probability \propto e^{\frac{- \epsilon |f(d) - r|}{2 \Delta q}}}$$

whereas the Laplace mechanism is defined as:

$$\mathcal{L}(d) := \text{choose r with probability \propto e^{\frac{- \epsilon |f(d) - r|}{\Delta f}}}$$

As I think that $$\Delta q$$ and $$\Delta f$$ are equal for this choice of function $$q$$, it seems that those two mechanisms are different. Where did I make a mistake ?

• Can you add a link to the paper you're referring to? – Ted Mar 3 at 21:09