Outer Bounds on the CEO Problem with Privacy Constraints

We investigate the rate-distortion-leakage region of the Chief Executive Officer (CEO) problem with a passive eavesdropper and privacy constraints, considering a general distortion measure. While an inner bound directly follows from the previous work, an outer bound is newly developed in this paper. To derive this bound, we introduce a new lemma tailored for analyzing privacy constraints. As a specific instance, we demonstrate that the tight bound for discrete and Gaussian sources is obtained when the eavesdropper can only observe the messages under logarithmic loss distortion. We further investigate the rate-distortion-leakage region for a scenario where the eavesdropper possesses the messages and side information under the same distortion, and provide an outer bound for this particular case. The derived outer bound differs from the inner bound by only a minor quantity that appears in the constraints associated with the privacy-leakage rates, and it becomes tight when the distortion is large.