Selection of Prior Weights for Weighted Model Averaging

This paper addresses the task of choosing prior weights for models that are to be used for weighted model averaging. Models that are very similar to each other should usually be given smaller weights than models that are quite distinct. Otherwise, the importance of a model in the weighted average could be increased by augmenting the set of models with duplicates of the model or virtual duplicates of it. Similarly, the importance of a particular model feature (a certain covariate, say) could be exaggerated by including many models with that feature. Ways of forming a correlation matrix that reflects the similarity between models are suggested. Then weighting schemes are proposed that assign prior weights to models on the basis of this matrix. The weighting schemes have a dilution property: smaller weights are given to models that are more highly correlated. Other desirable properties in a weighting scheme are identified and we examine the extent to which these properties are held by the proposed methods. Also, the weighting schemes are applied to real data sets and prior weights, posterior weights and Bayesian model averages are determined. For these data sets, empirical Bayes methods were used to form the correlation matrices that yield the prior weights. Predictive variances are examined as empirical Bayes methods can result in unrealistically small variances.