Asymptotically Minimax Prediction in Infinite Sequence Models

TL;DR

This paper develops asymptotically minimax predictive distributions for infinite sequence models, connecting them to function models, and demonstrates their optimality for ellipsoids and Sobolev spaces using Bayesian methods.

Contribution

It constructs and proves the asymptotic minimaxity of Bayesian predictive distributions for infinite sequence models with ellipsoid and Sobolev parameter spaces.

Findings

Bayesian predictive distribution with Gaussian prior is asymptotically minimax for ellipsoids.

Bayesian predictive distribution with Stein's priors is asymptotically minimax for Sobolev ellipsoids.

Efficient sampling method from the proposed Bayesian predictive distribution is provided.

Abstract

We study asymptotically minimax predictive distributions in an infinite sequence model. First, we discuss the connection between the prediction in the infinite sequence model and the prediction in the function model. Second, we construct an asymptotically minimax predictive distribution when the parameter space is a known ellipsoid. We show that the Bayesian predictive distribution based on the Gaussian prior distribution is asymptotically minimax in the ellipsoid. Third, we construct an asymptotically minimax predictive distribution for any Sobolev ellipsoid. We show that the Bayesian predictive distribution based on the product of Stein's priors is asymptotically minimax for any Sobolev ellipsoid. Finally, we present an efficient sampling method from the proposed Bayesian predictive distribution.