Identifying a distribution from its Gaussian-channel MMSE curve
For a real random variable $U$ with $\mathbb E[U^2] < \infty$, let $Z$ be a standard Gaussian independent of $U$ and let $$\operatorname{mmse}_U(s) = \mathbb E\bigl[(U - \mathbb E[U \mid \sqrt{s}\,U + Z])^2\bigr], \qquad s > 0,$$ be the minimum mean-square error of estimating $U$ from its observation through the Gaussian channel at signal-to-noise ratio $s$. The curve is unchanged when $U$ is replaced by $a + U$ or $a - U$ with $a \in \mathbb R$. Ledoux showed that two centred random variables that are determined by their moments, have nonnegative cumulants of all orders and have the same curve have the same law. Mansanarez, Poly and Swan proved that if $X$ is symmetric, determined by its moments and has positive cumulants of every even order, then every centred random variable with moments of all orders and the same curve has the law of $X$; the same holds when the even-order cumulants of $X$ alternate strictly in sign, for competitors whose fourth cumulant has the sign of that of $X$, which covers the uniform and Rademacher laws.
Problem. Let $X$ and $Y$ be real random variables with finite second moments such that $\operatorname{mmse}_X(s) = \operatorname{mmse}_Y(s)$ for every $s > 0$. Must $Y$ have the law of $a + \eta X$ for some $a \in \mathbb R$ and some $\eta \in \{-1, 1\}$?
Source: Guo, Wu, Shamai and Verdú (2011), Conjecture 1, for zero-mean random variables; see Ledoux (2016), §VI, and Mansanarez, Poly and Swan (2024), §2.2. Stated for random variables with finite second moments by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- IT, ST
- Keywords
- AIM Open Applied Problems, Gaussian channels, minimum mean-square error
References
@article{guo2011,
author = {Guo, Dongning and Wu, Yihong and Shamai (Shitz), Shlomo and Verd{\'u}, Sergio},
title = {Estimation in {G}aussian noise: properties of the minimum mean-square error},
journal = {IEEE Trans. Inform. Theory},
volume = {57},
number = {4},
year = {2011},
pages = {2371--2385},
doi = {10.1109/TIT.2011.2111010},
eprint = {1004.3332},
archivePrefix = {arXiv},
}
@article{ledoux2016,
author = {Ledoux, Michel},
title = {Heat flow derivatives and minimum mean-square error in {G}aussian noise},
journal = {IEEE Trans. Inform. Theory},
volume = {62},
number = {6},
year = {2016},
pages = {3401--3409},
doi = {10.1109/TIT.2016.2555809},
}
@article{mansanarez2024,
author = {Mansanarez, Paul and Poly, Guillaume and Swan, Yvik},
title = {Derivatives of entropy and the {MMSE} conjecture},
journal = {IEEE Trans. Inform. Theory},
volume = {70},
number = {11},
year = {2024},
pages = {7647--7663},
doi = {10.1109/TIT.2024.3466566},
eprint = {2311.04831},
archivePrefix = {arXiv},
}
@misc{aim-mmse-identification,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Identifying a distribution from its Gaussian-channel MMSE curve}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/563-mmse-identification.md},
note = {Entry 563 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00294, "Identifying a distribution from its Gaussian-channel MMSE curve", clopen.solutions.