  In this paper, we study the vector Gaussian Chief Executive Officer (CEO)
problem under logarithmic loss distortion measure. Specifically, $K \geq 2$
agents observe independently corrupted Gaussian noisy versions of a remote
vector Gaussian source, and communicate independently with a decoder or CEO
over rate-constrained noise-free links. The CEO wants to reconstruct the remote
source to within some prescribed distortion level where the incurred distortion
is measured under the logarithmic loss penalty criterion. We find an explicit
characterization of the rate-distortion region of this model. For the proof of
this result, we obtain an outer bound on the region of the vector Gaussian CEO
problem by means of a technique that relies on the de Bruijn identity and the
properties of Fisher information. The approach is similar to Ekrem-Ulukus outer
bounding technique for the vector Gaussian CEO problem under quadratic
distortion measure, for which it was there found generally non-tight; but it is
shown here to yield a complete characterization of the region for the case of
logarithmic loss measure. Also, we show that Gaussian test channels with
time-sharing exhaust the Berger-Tung inner bound, which is optimal.
Furthermore, we also show that the established result under logarithmic loss
provides an outer bound for a quadratic vector Gaussian CEO problem with
determinant constraint, for which we characterize the optimal rate-distortion
region.
