A Dimensionality Reduction Method for Finding Least Favorable Priors with a Focus on Bregman Divergence

TL;DR

This paper introduces a dimensionality reduction technique for efficiently finding least favorable priors in Bayesian estimation, focusing on Bregman divergences, enabling the use of gradient-based algorithms for minimax estimation.

Contribution

It proposes a novel method to reduce the infinite-dimensional optimization over priors to a finite-dimensional problem, facilitating practical computation of least favorable priors.

Findings

Dimensionality reduction bounds the problem to finite dimensions.

Enables use of projected gradient ascent for finding priors.

Applicable to a broad class of loss functions, specifically Bregman divergences.

Abstract

A common way of characterizing minimax estimators in point estimation is by moving the problem into the Bayesian estimation domain and finding a least favorable prior distribution. The Bayesian estimator induced by a least favorable prior, under mild conditions, is then known to be minimax. However, finding least favorable distributions can be challenging due to inherent optimization over the space of probability distributions, which is infinite-dimensional. This paper develops a dimensionality reduction method that allows us to move the optimization to a finite-dimensional setting with an explicit bound on the dimension. The benefit of this dimensionality reduction is that it permits the use of popular algorithms such as projected gradient ascent to find least favorable priors. Throughout the paper, in order to make progress on the problem, we restrict ourselves to Bayesian risks induced by a relatively large class of loss functions, namely Bregman divergences.