On Prior Selection and Covariate Shift of β-Bayesian Prediction Under α-Divergence Risk

ISSN： 0361-0926

Source： Communications in Statistics: Theory and Methods, Vol.39, Iss.8-9, 2010-01, pp. : 1655-1673

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Abstract

We investigate the prior selection problem for predicting an input-output relation by a generalized Bayesian method, α-Bayes prediction. The α-Bayes predictive distribution is given by minimizing the Bayes risk corresponding to the α-divergence that is a generalization of the Kullback-Leibler divergence. It is known that the effect of the prior to the performance of the usual Bayesian predictive distribution measured by the Kullback-Leibler divergence from the true distribution is asymptotically characterized by the Laplacian. We show that the α-divergence between the β-Bayes predictive distribution for next outputs and the true output distribution also has a similar characterization even if α might be different from β. We also investigate how the performance of the generalized Bayesian prediction behaves if the test and training input distributions are different.