On model selection by quasi-Bayesian predictive densities

M. Aitkin (Journal of the Royal Statistical Society B, vol. 53, p.111-42, 1991) has proposed an unorthodox Bayesian procedure for model selection based on quasi-Bayesian predictive densities. The idea is to use the data twice, once to obtain a prior density for the parameters, and the second time to analyze the data using the so obtained prior. Aitkin´s approach is discussed and compared with alternatives that do not violate the Bayesian paradigm. As a main alternative the authors consider the Bayesian predictive densities according to models and data. The important example of linear models is examined. Monte Carlo simulation results are provided that support the analysis. It is shown that model selections by quasi-Bayesian predictive densities yield poor results, particularly when the competing models are hierarchical.