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We study a class of $K$-encoder hypothesis testing against conditionalindependence problems. Under the criterion that stipulates minimization of theType II error subject to a (constant) upper bound $\epsilon$ on the Type Ierror, we characterize the set of encoding rates and exponent for both discretememoryless and memoryless vector Gaussian settings. For the DM setting, weprovide a converse proof and show that it is achieved using theQuantize-Bin-Test scheme of Rahman and Wagner. For the memoryless vectorGaussian setting, we develop a tight outer bound by means of a technique thatrelies on the de Bruijn identity and the properties of Fisher information. Inparticular, the result shows that for memoryless vector Gaussian sources therate-exponent region is exhausted using the Quantize-Bin-Test scheme with\textit{Gaussian} test channels; and there is \textit{no} loss in performancecaused by restricting the sensors' encoders not to employ time sharing.Furthermore, we also study a variant of the problem in which the source, notnecessarily Gaussian, has finite differential entropy and the sensors'observations noises under the null hypothesis are Gaussian. For this model, ourmain result is an upper bound on the exponent-rate function. The bound is shownto mirror a corresponding explicit lower bound, except that the lower boundinvolves the source power (variance) whereas the upper bound has the sourceentropy power. Part of the utility of the established bound is forinvestigating asymptotic exponent/rates and losses incurred by distributeddetection as function of the number of sensors.

Rate-Exponent Region for a Class of Distributed Hypothesis Testing Against Conditional Independence Problems