Asymptotic Properties of Bayesian Predictive Densities When the Distributions of Data and Target Variables are Different

作者：Komaki Fumiyasu^*

来源：Bayesian Analysis, 2015, 10(1): 31-51.

DOI：10.1214/14-BA886

• differential geometry
• Fisher-Rao metric
• Jeffreys prior
• Kullback-Leibler divergence
• predictive metric

Bayesian predictive densities when the observed data x and the target variable y to be predicted have different distributions are investigated by using the framework of information geometry. The performance of predictive densities is evaluated by the Kullback-Leibler divergence. The parametric models are formulated as Riemannian manifolds. In the conventional setting in which x and y have the same distribution, the Fisher-Rao metric and the Jeffreys prior play essential roles. In the present setting in which x and y have different distributions, a new metric, which we call the predictive metric, constructed by using the Fisher information matrices of x and y, and the volume element based on the predictive metric play the corresponding roles. It is shown that Bayesian predictive densities based on priors constructed by using non-constant positive superharmonic functions with respect to the predictive metric asymptotically dominate those based on the volume element prior of the predictive metric.

• 出版日期2015-3
• 单位RIKEN