If I have a pdf in the form $f(y|\theta,\phi)=\text{exp}\bigg(\frac{y\theta-b(\theta)}{a(\phi)}+c(y,\phi)\bigg)$, then $\theta$ is called the canonical parameter. I'm told we can get a link function $g$ from $\theta$ that links $E[Y]$ to $x^T\beta$, like so: $g(E[Y])=x^T\beta$.

E.g. For $Y\sim Poisson(\mu)$, $P(Y=y)=\exp(-\mu+y\log \mu-\log y!)$. From this, $\theta=\log\mu$. Hence, the link function is the log function: $\log E[Y]=x^T\beta$.

At the moment it seems like a total coincidence that the canonical parameter gives a function which just so happens to relate $E[Y]$ to a linear combination of some independent variables. Why does the canonical parameter have this property?



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