The least favorable noise

TL;DR

This paper investigates the worst-case additive noise perturbation that maximizes prediction error for a random variable, providing a characterization and solutions especially for infinitely divisible variables, and explores the impact of noise level on prediction accuracy.

Contribution

It characterizes the least favorable noise perturbation for prediction error, offering complete solutions for infinitely divisible variables and analyzing the effect of noise magnitude.

Findings

Characterization of least favorable perturbation

Complete solution for infinitely divisible variables

Noise level impacts prediction accuracy

Abstract

Suppose that a random variable $X$ of interest is observed perturbed by independent additive noise $Y$. This paper concerns the "the least favorable perturbation" $\hat Y_\ep$, which maximizes the prediction error $E(X-E(X|X+Y))^2$ in the class of $Y$ with $ \var (Y)\leq \ep$. We find a characterization of the answer to this question, and show by example that it can be surprisingly complicated. However, in the special case where $X$ is infinitely divisible, the solution is complete and simple. We also explore the conjecture that noisier $Y$ makes prediction worse.