Outlier Models and Prior Distributions in Bayesian Linear Regression

Abstract

SUMMARY Bayesian inference in regression models is considered using heavy-tailed error distri- butions to accommodate outliers. The particular class of distributions that can be con- structed as scale mixtures of normal distributions are examined and use is made of them as both error models and prior distributions in Bayesian linear modelling, includ- ing simple regression and more complex hierarchical models with structured priors depending on unknown hyperprior parameters. The modelling of outliers in nominally normal linear regression models using alternative error distributions which are heavy-tailed relative to the normal provides an automatic means of both detecting and accommodating possibly aberrant observations. Such realistic models do, however, often lead to analytically intractable analyses with complex posterior distributions in several dimensions that are difficult to summarize and understand. In this paper we consider a special yet rather wide class of heavy-tailed, unimodal and symmetric error distributions for which the analyses, though apparently intractable, can be examined in some depth by exploiting certain properties of the assumed error form. The distributions concerned are those that can be con- structed as scale mixtures of normal distributions. In his paper concerning location parameters, de Finetti (1961) discusses such distributions and suggests the hypothetical interpretation that "each observation is taken using an instrument with normal error, but each time chosen at random from a collection of instruments of different precisions, the distribution of the