# What makes the canonical link function special in GLMs? [duplicate]

Why is the canonical link function used so frequently with GLMs? What makes it "natural"?

Is there any reason to think that, $Q(\theta _i)$ (where $Q$ is the canonical link function, and $\theta _i$ is the parameter of interest) is better described by a linear combination of predictor variables than some other function of $\theta_i$.

That is, is there any reason to believe that:

$$Q(\theta _i) = \sum_j \beta _j x_{ij}$$

is superior to:

$$f(\theta _i) = \sum_j \beta _j x_{ij}$$ where $Q \neq f$

If not, is there anything else that makes $Q$ better than $f$? The text book I am using just mentions what a canonical link function is, and makes use of it (pretty much exclusively), but does not explain what distinguishes it from any arbitrary link function.

• Momo gives a good answer in this thread. Mar 24 '14 at 23:21
• Thanks for the link. A few things aren't quite making sense for me. Using Momo's notation: I understand that $\gamma '(\theta_i )= \mu_i$; for the binomial random variable $f(y_i;\theta_i)=(1-\theta_i) exp\{y_i ( log(\frac{\theta_i}{1-\theta_i})) \}$. My understanding was that for the binomial: $\gamma (\theta _i) =log(\frac{\theta_i}{1-\theta_i})$. If this is true, then $\gamma '(\theta_i )= \frac{1}{(\theta_i-1)^{2}} \neq E[Y_i] = \theta_i$. I guess I am off somewhere in my understanding. Mar 24 '14 at 23:59
• Correction: $\gamma '(\theta_i )= \frac{1}{\theta_i(1-\theta_i)} \neq E[Y_i] = \theta_i$. Mar 25 '14 at 0:05