Complete separation prevents a finite logistic MLE

Given: The design column $X=(-1,1{)}^{\mathsf{T}}$, responses $(0,1)$, the Bernoulli cumulant $b(\eta )=\log (1+e^{\eta })$, and $\beta \in \mathbb{R}$.

Generalized linear model with explicit conditioning: the canonical link is the choice $g=\tau$, equivalently ${\theta }_i={\eta }_i$.

Canonical-link score and Hessian: the canonical score is $X^{\mathsf{T}}(y-\mu )$ when $a(\varphi )=1$.

Strict concavity and uniqueness of a finite stationary maximizer: positive $b^{\mathrm{\prime }\mathrm{\prime }}$ and full column rank imply strict concavity, but the conclusion does not assert existence of a finite maximizer.

The Bernoulli-logit component mass on $\{0,1\}$ relative to counting measure is $f(y;\eta )=\exp \{y\eta -\log (1+e^{\eta })\}$. It sums to one; the support is fixed, the response is bounded, and its first two $\eta$ derivatives are locally bounded because $f^{\mathrm{\prime }}=f(y-p)$ and $f^{\mathrm{\prime }\mathrm{\prime }}=f\{(y-p{)}^2-p(1-p)\}$ with $p=e^{\eta }\mathrm{/}(1+e^{\eta })$. (Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials, The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t)