Asymptotic Properties of Bayesian Predictive Densities When the Distributions of Data and Target Variables are Different · arXivDesk
1503.07643Mar 26, 2015Published at http://dx.doi.org/10.1214/14-BA886 in the Bayesian Analysis (http://projecteuclid.org/euclid.ba) by the International Society of Bayesian Analysis (http://bayesian.org/)
Asymptotic Properties of Bayesian Predictive Densities When the Distributions of Data and Target Variables are Different
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
Nearby in the stack
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.