#gaussian-nll

A regression network predicts a mean $\mu(x)$ and a log-variance $s(x) = \log\sigma^2(x)$ and is trained with the Gaussian negative log-likelihood $\tfrac12\left[(y - \mu)^2 e^{-s} + s + \log 2\pi\right]$. For an example whose residual is $y - \mu = 3$, which predicted variance $\sigma^2 = e^{s}$ minimizes this loss?