= Finite maximum-likelihood estimate in a one-parameter logistic model
{title2=$\ell'(\beta)=\sum_i x_i[y_i-(1+e^{-\beta x_i})^{-1}]$}
For nonzero design entries the <Bernoulli logistic-regression model> has strictly concave log likelihood, since $-\ell''=\sum_i x_i^2\mu_i(1-\mu_i)>0$. A unique finite estimate exists exactly when its score is positive at minus infinity and negative at plus infinity. Perfect agreement of binary outcomes with the signs of the design can instead give a supremum approached at an infinite coefficient. If every design entry is zero, the coefficient is unidentifiable. <Newton method> uses the score divided by this observed information, with damping when needed.
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