Consider a generalized linear model for independent observations $Y_{1}, \ldots, Y_{n}$, with $\mathbb{E}\left(Y_{i}\right)=\mu_{i}$ for $i=1, \ldots, n$. What is a linear predictor? What is meant by the link function? If $Y_{i}$ has model function (or density) of the form

$$f\left(y_{i} ; \mu_{i}, \sigma^{2}\right)=\exp \left[\frac{1}{\sigma^{2}}\left\{\theta\left(\mu_{i}\right) y_{i}-K\left(\theta\left(\mu_{i}\right)\right)\right\}\right] a\left(\sigma^{2}, y_{i}\right)$$

for $y_{i} \in \mathcal{Y} \subseteq \mathbb{R}, \mu_{i} \in \mathcal{M} \subseteq \mathbb{R}, \sigma^{2} \in \Phi \subseteq(0, \infty)$, where $a\left(\sigma^{2}, y_{i}\right)$ is a known positive function, define the canonical link function.

Now suppose that $Y_{1}, \ldots, Y_{n}$ are independent with $Y_{i} \sim \operatorname{Bin}\left(1, \mu_{i}\right)$ for $i=1, \ldots, n$. Derive the canonical link function.