= Concavity of the Poisson regression likelihood
For a full-column-rank design $X$ and positive fitted means $\mu_i=\exp(x_i^T\beta)$, the <log-likelihood> has Hessian $-X^T\operatorname{diag}(\mu_i)X$, which is negative definite. Thus any finite stationary point is the unique <maximum-likelihood estimator>. Full rank does not guarantee existence: with an intercept and all counts zero, the supremum is approached as the intercept tends to $-\infty$. If every observed count is positive, a full-rank design does give a finite maximum: each function $y_i\eta_i-e^{\eta_i}$ tends to $-\infty$ in either tail and is bounded above, and $\|X\beta\|\to\infty$ as $\|\beta\|\to\infty$, so the negative <log-likelihood> is coercive.
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