(a) For a given model with likelihood $L(\beta), \beta \in \mathbb{R}^{p}$, define the Fisher information matrix in terms of the Hessian of the log-likelihood.

Consider a generalised linear model with design matrix $X \in \mathbb{R}^{n \times p}$, output variables $y \in \mathbb{R}^{n}$, a bijective link function, mean parameters $\mu=\left(\mu_{1}, \ldots, \mu_{n}\right)$ and dispersion parameters $\sigma_{1}^{2}=\ldots=\sigma_{n}^{2}=\sigma^{2}$. Assume $\sigma^{2}$ is known.

(b) State the form of the log-likelihood.

(c) For the canonical link, show that when the parameter $\sigma^{2}$ is known, the Fisher information matrix is equal to

$$\sigma^{-2} X^{T} W X$$

for a diagonal matrix $W$ depending on the means $\mu$. Identify $W$.