# Uncertainty Principles in Risk-Aware Statistical Estimation

arXiv: https://arxiv.org/abs/2104.14283 · HTML: https://mlsanity.com/paper/2104.14283/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 20 references that are in the map.
- First posted on arXiv 29 Apr 2021.

## Builds on (most cited first, 11 of 11)

- [Fairness risk measures](https://mlsanity.com/paper/1901.08665/) (2019)
- [Adaptive Sampling for Stochastic Risk-Averse Learning](https://mlsanity.com/paper/1910.12511/) (2019)
- [Ultra-Reliable and Low-Latency Wireless Communication: Tail, Risk and Scale](https://mlsanity.com/paper/1801.01270/) (2018)
- [A Risk-Sensitive Finite-Time Reachability Approach for Safety of Stochastic Dynamic Systems](https://mlsanity.com/paper/1902.11277/) (2019)
- [Risk-Averse Stochastic Convex Bandit](https://mlsanity.com/paper/1810.00737/) (2018)
- [Fast Risk Assessment for Autonomous Vehicles Using Learned Models of Agent Futures](https://mlsanity.com/paper/2005.13458/) (2020)
- [Safety-Aware Optimal Control of Stochastic Systems Using Conditional Value-at-Risk](https://mlsanity.com/paper/1802.07903/) (2018)
- [Bi-directional Value Learning for Risk-aware Planning Under Uncertainty: Extended Version](https://mlsanity.com/paper/1902.05698/) (2019)
- [Noisy Linear Convergence of Stochastic Gradient Descent for CV@R Statistical Learning under Polyak-Lojasiewicz Conditions](https://mlsanity.com/paper/2012.07785/) (2020)
- [A Stochastic Subgradient Method for Distributionally Robust Non-Convex Learning](https://mlsanity.com/paper/2006.04873/) (2020)
- [Risk-Aware MMSE Estimation](https://mlsanity.com/paper/1912.02933/) (2019)

## Abstract (arXiv)

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.