Uncertainty Principles in Risk-Aware Statistical Estimation

TL;DR

This paper introduces a new uncertainty principle in risk-aware statistical estimation, establishing a fundamental trade-off between mean squared error and predictive variance, with implications for understanding estimator performance.

Contribution

It formulates a novel uncertainty principle linking MSE and risk, related to the Pareto frontier of constrained estimation, and connects this to a new measure of distribution skewness.

Findings

The product of MSE and risk is bounded below by a computable constant.

The constant relates to a new topological measure of distribution skewness.

Numerical simulations illustrate the theoretical results.

Abstract

We present a new uncertainty principle for risk-aware statistical estimation, effectively quantifying the inherent trade-off between mean squared error ($\mse$) and risk, the latter measured by the associated average predictive squared error variance ($\sev$), for every admissible estimator of choice. Our uncertainty principle has a familiar form and resembles fundamental and classical results arising in several other areas, such as the Heisenberg principle in statistical and quantum mechanics, and the Gabor limit (time-scale trade-offs) in harmonic analysis. In particular, we prove that, provided a joint generative model of states and observables, the product between $\mse$ and $\sev$ is bounded from below by a computable model-dependent constant, which is explicitly related to the Pareto frontier of a recently studied $\sev$-constrained minimum $\mse$ (MMSE) estimation problem. Further, we show that the aforementioned constant is inherently connected to an intuitive new and rigorously topologically grounded statistical measure of distribution skewness in multiple dimensions, consistent with Pearson's moment coefficient of skewness for variables on the line. Our results are also illustrated via numerical simulations.