Finite maximum-likelihood estimate in a one-parameter logistic model (${\ell }^{\prime }(\beta )={\sum }_i{x}_i[y_i-(1+e^{-\beta x_i}{)}^{-1}]$)

For nonzero design entries the Bernoulli logistic-regression model has strictly concave log likelihood, since $-{\ell }^{\prime \prime }={\sum }_i{x}_i^2{\mu }_i(1-{\mu }_i)>0$. 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.