  A new Bayesian method for performing an image domain search for line-emitting
galaxies is presented. The method uses both spatial and spectral information to
robustly determine the source properties, employing either simple Gaussian, or
other physically motivated models whilst using the evidence to determine the
probability that the source is real. In this paper, we describe the method, and
its application to both a simulated data set, and a blind survey for cold
molecular gas using observations of the Hubble Deep Field North taken with the
Plateau de Bure Interferometer. We make a total of 6 robust detections in the
survey, 5 of which have counterparts in other observing bands. We identify the
most secure detections found in a previous investigation, while finding one new
probable line source with an optical ID not seen in the previous analysis. This
study acts as a pilot application of Bayesian statistics to future searches to
be carried out both for low-$J$ CO transitions of high redshift galaxies using
the JVLA, and at millimeter wavelengths with ALMA, enabling the inference of
robust scientific conclusions about the history of the molecular gas properties
of star-forming galaxies in the Universe through cosmic time.
