Robust Bayesian regression in astronomy

Model mis-specification (e.g. the presence of outliers) is commonly encountered in astronomical analyses, often requiring the use of ad hoc algorithms (e.g. sigma-clipping). We develop and implement a generic Bayesian approach to linear regression, based on Student's t-distributions, that is robust to outliers and mis-specification of the noise model. Our method is validated using simulated datasets with various degrees of model mis-specification; the derived constraints are shown to be systematically less biased than those from a similar model using normal distributions. We demonstrate that, for a dataset without outliers, a worst-case inference using t-distributions would give unbiased results with $\lesssim\!10$ per cent increase in the reported parameter uncertainties. We also compare with existing analyses of real-world datasets, finding qualitatively different results where normal distributions have been used and agreement where more robust methods have been applied. A Python implementation of this model, t-cup, is made available for others to use.