In a GLM, if the response variable has a Gamma distribution, why is the inverse used as the link function, i.e.: $\mu = -(X\beta)^{-1}$?
In particular, why is the inverse the canonical link? Does it have to do with the natural parameters of the gamma distribution?
why is the inverse used as the link function, i.e.: $μ=−(Xβ)^{−1}$
That's actually the mean-function $\mu(\eta)$. The link function is $\eta(\mu)$. However, both are in the form of a negative reciprocal in this case, since the negative of the reciprocal is its own inverse-function.
In particular, why is the inverse the canonical link? Does it have to do with the natural parameters of the gamma distribution?
Yes, that's where canonical links come from.
See for example, the Wikipedia article on the Generalized Linear model, in the section on the link function:
When using a distribution function with a canonical parameter $\theta$, the canonical link function is the function that expresses $\theta$ in terms of $\mu$
('canonical parameter' being another term for natural parameter)
