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.
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