Canonical-link score and Hessian

Given: A finite fixed design $X$, an admissible $\beta$, the common-scale canonical-link GLM, and an observed response vector whose component densities are positive at the corresponding canonical parameters.

Generalized linear model with explicit conditioning: ${\eta }_i=x_i^{\mathsf{T}}\beta$, the canonical link is ${\theta }_i={\eta }_i$, and the common-scale log likelihood is the finite sum of component log densities.

The product response law has the product density: the joint density is the product of component densities, so wherever all factors are positive its log likelihood is their sum.

Generalized linear model with explicit conditioning: for the canonical link, ${\theta }_i={\eta }_i=\tau ({\mu }_i)$ and $b^{\mathrm{\prime }}(\tau ({\mu }_i))={\mu }_i$.

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$: for differentiable scalar functions, $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$ under the stated limit-point hypotheses.

Matrix differentials obey the sum rule, product rule, and adjoint rule: in the real Frechet sense, $d(AB)=dAB+AdB$ and $d(A+B)=dA+dB$.