Optimality of Right-Invariant Priors

TL;DR

This paper establishes measure-theoretic conditions under which right-invariant priors are optimal for prediction in models with group symmetry, with applications to multivariate normals and Gaussian processes.

Contribution

It provides a measure-theoretic proof of the optimality of right-invariant priors under regularity conditions, extending previous results.

Findings

Right-invariant priors minimize worst-case Kullback-Leibler risk in certain models.

Strong optimality results for multivariate normal and Gaussian process predictions.

Numerical comparisons show advantages of right-invariant priors over other methods.

Abstract

We discuss optimal prediction for families of probability distributions with a locally compact topological group structure. Right-invariant priors were previously shown to yield a posterior predictive distribution minimizing the worst-case Kullback-Leibler risk among all predictive procedures. However, the assumptions for the proof are so strong that they rarely hold in practice and it is unclear when the density functions used in the proof exist. Therefore, we provide a measure-theoretic proof, establishing adequate regularity assumptions. As applications, we show a strong optimality result for next-sample prediction for multivariate normal distributions and Gaussian Process regression with fixed lengthscale. We also discuss uniqueness and numerically evaluate prediction with right-invariant priors against other objective priors and plug-in prediction.