# An MMSE Lower Bound via Poincar'e Inequality

arXiv: https://arxiv.org/abs/2205.05848 · HTML: https://mlsanity.com/paper/2205.05848/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 39 references that are in the map.
- First posted on arXiv 12 May 2022.

## Builds on (most cited first, 1 of 1)

- [A General Derivative Identity for the Conditional Mean Estimator in Gaussian Noise and Some Applications](https://mlsanity.com/paper/2104.01883/) (2021)

## Built on by (most cited first, 1 of 1)

- [Lower Bounds on the MMSE of Adversarially Inferring Sensitive Features](https://mlsanity.com/paper/2505.09004/) (2025)

## Abstract (arXiv)

This paper studies the minimum mean squared error (MMSE) of estimating X ∈ ℝd from the noisy observation Y ∈ ℝk, under the assumption that the noise (i.e., Y|X) is a member of the exponential family. The paper provides a new lower bound on the MMSE. Towards this end, an alternative representation of the MMSE is first presented, which is argued to be useful in deriving closed-form expressions for the MMSE. This new representation is then used together with the Poincar'e inequality to provide a new lower bound on the MMSE. Unlike, for example, the Cram'er-Rao bound, the new bound holds for all possible distributions on the input X. Moreover, the lower bound is shown to be tight in the high-noise regime for the Gaussian noise setting under the assumption that X is sub-Gaussian. Finally, several numerical examples are shown which demonstrate that the bound performs well in all noise regimes.