Bayesian analysis of outliers via akaike's predictive likelihood of a model

A set of independent observations is assumed to come from one or more normal populations having the same unknown variance and different unknown means. Ignorance priors are associated with these parameters. The number of populations is also unknown as is the number of observations from each, but priors are chosen for these quantities which make it very likely that one population is dominant. Observations from the rest are considered outliers. Using these priors in conjunction with Akaike's predictive likelihood, which is derived for the class of models considered, one can obtain a quasi-Bayesian posterior probability for each possible model. A “robust” estimate of the mean value of the dominant population and “corrected” values for the outliers can be calculated from the posterior probabilities, once the outliers have been designated. Darwin's data and Herndon's data are analyzed to illustrate the procedure.