Abstract
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.
Highlights
Let x(N ) = (x1, · · ·, xN ) be independent N data distributed according to a probability density, p(x|θ), that belongs to a d-dimensional parametric model, {p(x|θ) : θ ∈ Θ}, where θ = (θ1, · · ·, θd ) is an unknown d-dimensional parameter and Θ is the parameter space
We focus on a criterion of constructing minimax predictive densities under the Kullback–Leibler risk
We consider the settings where there exists a unique maximum point of the trace, X,ij g (θ)gijY (θ); for example, these settings appear in predictions based on the binary regression model, where the covariates of the data and the target variables are not identical
Summary
We consider the settings where there exists a unique maximum point of the trace, X,ij g (θ)gijY (θ); for example, these settings appear in predictions based on the binary regression model, where the covariates of the data and the target variables are not identical. When there exists a unique maximum point of the trace, g X,ij (θ)gijY (θ), we construct the asymptotically constant-risk prior, π(θ; N ), up to O(N −2 ), by making the prior dependent on the sample size, N , as:. We clarify the subminimax estimator problem based on the mean squared error from the viewpoint of the prediction where the distributions of data and target variables are different and have a common unknown parameter. Any relationship between such subminimax estimator problems and predictions have not been investigated, and further, in general, the improvement by the minimax estimator over the subminimax estimators has not been investigated
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