I am studying GLMs for exponential families and wondering how the density of exponential families gives rise to a canonical link function.
Following the notation from Fahrmeier (Regression), the density of an exponential family is
$$f(y \mid \theta) = \text{exp}\big (\frac{y \theta - b(\theta)}{\Phi}w + c(y,\Phi,w) \big ).$$
In this notation, $\mu = E[Y] = b'(\theta)$.
Also, in typical GLM notation, one often writes $g(E[Y \mid X]) = X \beta$, or $E[Y \mid X] = g^{-1}(X \beta)$.
My question: How do I automatically get a canonical link for each distribution in the exponential family? If $b'$ were invertible and $\theta \in \mathbb{R}$, then I could simply set $g = (b')^{-1}$ and try to predict $\theta$ with the linear predictor. But why is $b'$ always invertible and $\theta \in \mathbb{R}$? Many sources just tend to write down the parameter mapping $\theta(\mu)$ and it's inverse without stating explicit assumptions. Thanks!
