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We present a method to simultaneously model the dust far-infrared spectral energy distribution (SED) and the total infrared $-$ carbon monoxide (CO) integrated intensity $(S_{\rm IR}-I_{\rm CO})$ relationship. The modelling employs a hierarchical Bayesian (HB) technique to estimate the dust surface density, temperature ($T_{\rm eff}$), and spectral index at each pixel from the observed far-infrared (FIR) maps. Additionally, given the corresponding CO map, the method simultaneously estimates the slope and intercept between the FIR and CO intensities, which are global properties of the observed source. The model accounts for correlated and uncorrelated uncertainties, such as those present in Herschel observations. Using synthetic datasets, we demonstrate the accuracy of the HB method, and contrast the results with common non-hierarchical fitting methods. As an initial application, we model the dust and gas on 100 pc scales in the Magellanic Clouds from Herschel FIR and NANTEN CO observations. The slopes of the $\log S_{\rm IR}-\log I_{\rm CO}$ relationship are similar in both galaxies, falling in the range 1.1$-$1.7. However, in the SMC the intercept is nearly 3 times higher, which can be explained by its lower metallicity than the LMC, resulting in a larger $S_{\rm IR}$ per unit $I_{\rm CO}$. The HB modelling evidences an increase in $T_{\rm eff}$ in regions with the highest $I_{\rm CO}$ in the LMC. This may be due to enhanced dust heating in the densest molecular regions from young stars. Such simultaneous dust and gas modelling may reveal variations in the properties of the ISM and its association with other galactic characteristics, such as star formation rates and/or metallicities., Comment: 17 pages, 14 figures, Updated to match MNRAS accepted version

We estimate the parameters of the Kennicutt-Schmidt (KS) relationship, linking the star formation rate (Sigma_SFR) to the molecular gas surface density (Sigma_mol), in the STING sample of nearby disk galaxies using a hierarchical Bayesian method. This method rigorously treats measurement uncertainties, and provides accurate parameter estimates for both individual galaxies and the entire population. Assuming standard conversion factors to estimate Sigma_SFR and Sigma_mol from the observations, we find that the KS parameters vary between galaxies, indicating that no universal relationship holds for all galaxies. The KS slope of the whole population is 0.76, with the 2sigma range extending from 0.58 to 0.94. These results imply that the molecular gas depletion time is not constant, but varies from galaxy to galaxy, and increases with the molecular gas surface density. Therefore, other galactic properties besides just Sigma_mol affect Sigma_SFR, such as the gas fraction or stellar mass. The non-universality of the KS relationship indicates that a comprehensive theory of star formation must take into account additional physical processes that may vary from galaxy to galaxy., Comment: 7 pages, 2 figures, 1 table. Updated to match MNRAS accepted version