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We present a method to simultaneously model the dust far-infrared spectralenergy distribution (SED) and the total infrared $-$ carbon monoxide (CO)integrated intensity $(S_{\rm IR}-I_{\rm CO})$ relationship. The modellingemploys a hierarchical Bayesian (HB) technique to estimate the dust surfacedensity, temperature ($T_{\rm eff}$), and spectral index at each pixel from theobserved far-infrared (FIR) maps. Additionally, given the corresponding CO map,the method simultaneously estimates the slope and intercept between the FIR andCO intensities, which are global properties of the observed source. The modelaccounts for correlated and uncorrelated uncertainties, such as those presentin Herschel observations. Using synthetic datasets, we demonstrate the accuracyof the HB method, and contrast the results with common non-hierarchical fittingmethods. As an initial application, we model the dust and gas on 100 pc scalesin the Magellanic Clouds from Herschel FIR and NANTEN CO observations. Theslopes of the $\log S_{\rm IR}-\log I_{\rm CO}$ relationship are similar inboth galaxies, falling in the range 1.1$-$1.7. However, in the SMC theintercept is nearly 3 times higher, which can be explained by its lowermetallicity than the LMC, resulting in a larger $S_{\rm IR}$ per unit $I_{\rmCO}$. The HB modelling evidences an increase in $T_{\rm eff}$ in regions withthe highest $I_{\rm CO}$ in the LMC. This may be due to enhanced dust heatingin the densest molecular regions from young stars. Such simultaneous dust andgas modelling may reveal variations in the properties of the ISM and itsassociation with other galactic characteristics, such as star formation ratesand/or metallicities.

Simultaneously modelling far-infrared dust emission and its relation to CO emission in star forming galaxies