  We study a class of $K$-encoder hypothesis testing against conditional
independence problems. Under the criterion that stipulates minimization of the
Type II error subject to a (constant) upper bound $\epsilon$ on the Type I
error, we characterize the set of encoding rates and exponent for both discrete
memoryless and memoryless vector Gaussian settings. For the DM setting, we
provide a converse proof and show that it is achieved using the
Quantize-Bin-Test scheme of Rahman and Wagner. For the memoryless vector
Gaussian setting, we develop a tight outer bound by means of a technique that
relies on the de Bruijn identity and the properties of Fisher information. In
particular, the result shows that for memoryless vector Gaussian sources the
rate-exponent region is exhausted using the Quantize-Bin-Test scheme with
\textit{Gaussian} test channels; and there is \textit{no} loss in performance
caused by restricting the sensors' encoders not to employ time sharing.
Furthermore, we also study a variant of the problem in which the source, not
necessarily Gaussian, has finite differential entropy and the sensors'
observations noises under the null hypothesis are Gaussian. For this model, our
main result is an upper bound on the exponent-rate function. The bound is shown
to mirror a corresponding explicit lower bound, except that the lower bound
involves the source power (variance) whereas the upper bound has the source
entropy power. Part of the utility of the established bound is for
investigating asymptotic exponent/rates and losses incurred by distributed
detection as function of the number of sensors.
