"The canonical link of Gamma GLM is $g(x)=1/x$ is often not very practical. Log-link is more appropriated in most cases." One reason I can think of is that log-link makes sure $\mu$, the mean, is always greater than zero. (please correct me if I'm wrong.) But I believe there are other considerations.
So why people often choose the log-link over the canonical one? Is it related to the fact that $\log$ transformation stabilizes data variance?
A good question. I see the following reasons:
A log link produces a multiplicative model on the original scale and is thus easy to interprete. This is not true for the canonical link of the Gamma GLM.
In insurance pricing, the expected claim amount $E(L)$ is often decomposed into $E(L) = E(F)E(S)$, where $F$ is the claim frequency and $S$ the claim height. If both $E(F)$ and $E(S)$ are modeled by GLMs with log link, the coefficients of both models can be easily combined to determine the effects on the expected loss $E(L)$.
Not using the canonical link introduces a prediction bias on the original scale, i.e. the average prediction differs from the average response. This would be a reason against the log link. However, in my experience, this bias is small with the log link and can easily be fixed by a multiplicative correction factor.
