Selection of Weights for Weighted Model Averaging

Abstract

We address the task of choosing prior weights for models that are to be used for weighted model averaging. Models that are very similar 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 give smaller weights to models that are more highly correlated. Other desirable properties of a weighting scheme are identified, and we examine the extent to which these properties are held by the proposed methods. The weighting schemes are applied to real data, and prior weights, posterior weights and Bayesian model averages are determined. For these data, 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.

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“…When there is no information to support explicit weight values for each model in a collection, the use of constant weights has been suggested (Bailer et al, 2005;Faes et al, 2007;Wheeler & Bailer, 2008). A third alternative, generally used here, assumes initially that all models are equally likely a priori, and then adjusts the weights as described by Garthwaite and Mubwandarikwa (2010) [GM in what follows] and by Garthwaite, Critchley, Anaya-Izquierdo, and Mubwandarikwa (2012) to reduce the weights of models that produce highly correlated predictions, which otherwise could lead to inappropriate a priori weights (Clyde & George, 2004).…”