We investigate the so-called "MMSE conjecture" from Guo et al. (2011) which asserts that two distributions on the real line with the same entropy along the heat flow coincide up to translation and symmetry. Our approach follows the path breaking contribution Ledoux (1995) which gave algebraic representations of the derivatives of said entropy in terms of multivariate polynomials. The main contributions in this note are (i) we obtain the leading terms in the polynomials from Ledoux (1995), and (ii) we provide new conditions on the source distributions ensuring the MMSE conjecture holds. As illustrating examples, our findings cover the cases of uniform and Rademacher distributions, for which previous results in the literature were inapplicable.

翻译：本文研究了Guo等人(2011)提出的所谓"MMSE猜想"，该猜想断言：两条实线上在热流作用下具有相同熵的分布，除了平移和对称变换外完全一致。我们的方法沿袭了Ledoux(1995)的开创性工作，该工作利用多元多项式给出了熵导数的代数表示。本文的主要贡献在于：(i) 获得了Ledoux(1995)多项式中前导项的显式表达；(ii) 提出了确保MMSE猜想成立的新源分布条件。作为示例说明，我们的结论涵盖了均匀分布和Rademacher分布的情形，而此前文献中的已有结果无法适用于这些情形。