Differentially Private Policy Evaluation ( Supplementary Material )

Proof. Fix a pair of neighbouring datasets X ' X ′ and let E ⊆ R be any measurable set. Let ΘX,X′ be as in the statement and write ΘX,X′ = R \ΘX,X′ . Using the assumptions on ΘX,X′ we see that P[θX ∈ E] = P[θX ∈ E ∩ΘX,X′ ] + P[θX ∈ E ∩ΘX,X′ ] ≤ eP[θX′ ∈ E ∩ΘX,X′ ] + δ ≤ eP[θX′ ∈ E] + δ . Now we proceed with the proof of Lemma 1. Let X ' X ′ be two neighbouring datasets and let us write Z1 = ZX and Z2 = ZX′ for simplicity. Thus, for i = 1, 2 we have that Zi ∼ N (μi, σ i I) are d-dimensional independent Gaussian random variables whose means and variances satisfy the assumptions of Lemma 1 for some ε, δ > 0. The density function of Zi is denoted by fZi(z). In order to be able to apply Lemma 2 we want to show that the privacy loss between Z1 and Z2 defined as L(z) = ln fZ1(z) fZ2(z) (1)