Asymptotically Constant-Risk Predictive Densities When the Distributions of Data and Target Variables Are Different
by
Keisuke Yano 1,* and Fumiyasu Komaki 1,2
Keisuke Yano 1,* and Fumiyasu Komaki 1,2
1
Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
2
RIKEN Brain Science Institute, 2-1 Hirosawa, Wako City, Saitama 351-0198, Japan
*
Author to whom correspondence should be addressed.
Entropy 2014, 16(6), 3026-3048; https://doi.org/10.3390/e16063026
Received: 28 March 2014 / Revised: 9 May 2014 / Accepted: 22 May 2014 / Published: 28 May 2014
(This article belongs to the Special Issue Information Geometry)
We investigate the asymptotic construction of constant-risk Bayesian predictive densities under the Kullback–Leibler risk when the distributions of data and target variables are different and have a common unknown parameter. It is known that the Kullback–Leibler risk is asymptotically equal to a trace of the product of two matrices: the inverse of the Fisher information matrix for the data and the Fisher information matrix for the target variables. We assume that the trace has a unique maximum point with respect to the parameter. We construct asymptotically constant-risk Bayesian predictive densities using a prior depending on the sample size. Further, we apply the theory to the subminimax estimator problem and the prediction based on the binary regression model.
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Keywords:
Bayesian prediction; Fisher information; Kullback–Leibler divergence; minimax; predictive metric; subminimax estimator
This is an open access article distributed under the Creative Commons Attribution License
MDPI and ACS Style
Yano, K.; Komaki, F. Asymptotically Constant-Risk Predictive Densities When the Distributions of Data and Target Variables Are Different. Entropy 2014, 16, 3026-3048.
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