On the lack of invariance of some Bayesian outlier models

The property of non-invariance to the scale of the data for the Abraham and Box (1978) Bayesian outlier model, and the model that generalizes the Guttman, Dutter and Freeman (1978) outlier analysis is discussed. This drawback is due to the non-informative prior taken for the parameters. Freeman (1980) expected that most posterior weight would be put on the model with the most outliers if the improper prior is used. We show that this may not be correct. An illustrative example is given.