  Transient radio signals of astrophysical origin present an avenue for
studying the dynamic universe. With the next generation of radio
interferometers being planned and built, there is great potential for detecting
and studying large samples of radio transients. Currently-used image-based
techniques for detecting radio sources have not been demonstrated to be
optimal, and there is a need for development of more sophisticated algorithms,
and methodology for comparing different detection techniques. A
visibility-space detector benefits from our good understanding of
visibility-space noise properties, and does not suffer from the image artifacts
and need for deconvolution in image-space detectors. In this paper, we propose
a method for designing optimal source detectors using visibility data, building
on statistical decision theory. The approach is substantially different to
conventional radio astronomy source detection. Optimal detection requires an
accurate model for the data, and we present a realistic model for the
likelihood function of radio interferometric data, including the effects of
calibration, signal confusion and atmospheric phase fluctuations. As part of
this process, we derive fundamental limits on the calibration of an
interferometric array, including the case where many relatively weak "in-beam"
calibrators are used. These limits are then applied, along with a model for
atmospheric phase fluctuations, to determine the limits on measuring source
position, flux density and spectral index, in the general case. We then present
an optimal visibility-space detector using realistic models for an
interferometer.
