Bayesian analysis of the linear model with heterogeneous variance

Linear models are among the most common statistical tools in forestry, and indeed in all science. It is widely known that when linear models are applied to attributes of size (e.g., individual tree volumes), the conditional variance may be heterogeneous. Under these circumstances, the usual least squares estimator remains consistent, but is no longer efficient. This has been appreciated in forestry for some time, and various solutions have been recommended over the years. In this paper, we propose a Bayesian solution. Bayesians would prefer this method to previous solutions for many reasons. However, even non-Bayesians may wish to consider the method as it yields (as will be shown) solutions quite close to the maximum likelihood solution, along with the marginal posterior distribution of each parameter.