  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 also has its own Gaussian noisy
observation of the source and 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. The result can be
seen as the counterpart to the vector Gaussian setting of that by
Courtade-Weissman which provides the rate-distortion region of the model in the
discrete memoryless setting. For the proof of this result, we obtain an outer
bound 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, application of our results allows us to find the complete
solutions of two related problems: a quadratic vector Gaussian CEO problem with
determinant constraint and the vector Gaussian distributed Information
Bottleneck problem. Finally, we develop Blahut-Arimoto type algorithms that
allow to compute numerically the regions provided in this paper, for both
discrete and Gaussian models. We illustrate the efficiency of our algorithms
through some numerical examples.
