Heat Flow Derivatives and Minimum Mean-Square Error in Gaussian Noise

We connect recent developments on Gaussian noise estimation and the Minimum Mean-Square Error to earlier results on entropy and Fisher information heat flow expansion. In particular, the derivatives of the Minimum mean-square error with respect to the noise parameter are related to the heat flow derivatives of the Fisher information and a special Lie algebra structure on iterated gradients. The results lead in particular to a partial answer to the Minimum mean-square error conjecture.