The IRAS discovery of infrared cirrus clouds was the starting point of correlation studies of neutral atomic hydrogen and dust far-infrared emission. These yield important insights into the gas and dust physics, the accretion history of the Milky Way, the X_CO factor, and the distribution of the CO-dark molecular gas. So far, full-sky analyses were limited by the angular resolution of the HI data of about one degree. The recently finished Effelsberg-Bonn HI Survey (EBHIS) improved this limit by an order of magnitude. Here, we aim to consistently quantify the relation between dust and gas, using the most accurate full-sky data sets. Moreover, a Bayesian framework and state-of-the-art methods of inference and image analyses allow us to deduce all-sky maps of e.g. the X_CO factor, dust emissivity, (CO-dark) molecular gas and the Cosmic Infrared Background. We present the first results of this work and compare our results with targeted observations.