Bayesian measures of surprise for outlier detection

From a Bayesian point of view, testing whether an observation is an outlier is usually reduced to a testing problem concerning a parameter of a contaminating distribution. This requires elicitation of both (i) the contaminating distribution that generates the outlier and (ii) prior distributions on its parameters. However, very little information is typically available about how the possible outlier could have been generated. Thus easy, preliminary checks in which these assessments can often be avoided may prove useful. Several such measures of surprise are derived for outlier detection in normal models. Results are applied to several examples. Default Bayes factors, where the contaminating model is assessed but not the prior distribution, are also computed.