This is loosely tied to the answer here
In GLMs we model the natural parameter (assume we’re using the canonical link) using a linear predictor. What supports this assumption of linearity?
Couldn’t the natural parameter be related to explanatory variables in a non-linear manner? If so, why does that not seem to be a major concern for practitioners using GLMs?
In linking to Momo's answer I presume you're specifically asking about the comment right at the end:
Note however, that there is no a priori reason why the effects in the model should be additive on the scale given by this or any other link.
This is not particularly different from the same question applied to regression. Since that context is simpler, let's start there.
In linear regression you have $E(Y|x_1,x_2,...x_n)$ is linear in the supplied predictors $(1,x_1,x_2,...x_n)$. However, this may encompass curved relationships, just as it can in GLMs, because the supplied predictors are not limited to the original features, and $Y$ itself may (at least in some situations) be transformed.
The question remains, why should it be that $E(Y|X) = X\beta$?
The obvious answer is it needn't be the case, and there are tools for dealing with the cases where it isn't. For example, we have nonlinear regression, which deals with considerably more general problems of the form $E(Y|X) = f(X,\beta)$.
Nevertheless, in many cases an additive model on a suitable scale is a reasonable approximation, and its simplicity in those circumstances can have a number of advantages.
Indeed, even in nonlinear regression, the model is estimated by building a sequence of linear regressions, which attempt to approximate the behavior of the nonlinear model in a sequence of places near the least squares solution; clearly linear models can be a suitable approximation, at least locally, in the right situation.
In GLMs we model the natural parameter (assume we’re using the canonical link) using a linear predictor. What supports this assumption of linearity?
Firstly note that we're not limited to the canonical link. There's multiple link functions in common use for most of the commonly used models. But, as Momo points out, why should it be linear under any link?
One thing to note is that relationships that are not additive under one link may be nearly additive under another; going back to the regression side, that's the premise of methods like ACE and AVAS, which attempt to find a transformation of $Y$ such that the relationship will be approximately linear in transformed $x$'s. This is often the case. Indeed, there's a connection to the Kolmogorov-Arnold representation theorem here you can then think of a choice of link functions and transformation of predictors as an attempt to obtain the best single-transformation approximation of such a relationship.
Couldn’t the natural parameter be related to explanatory variables in a non-linear manner? If so, why does that not seem to be a major concern for practitioners using GLMs?
It is a concern; many practitioners do attempt to assess whether their models capture the relationships between the variables, or fail to through a variety of tools including theory, diagnostics, and so forth.
If generalized additive models (even with other choices of link) cannot capture the relationship, there's other possibilities (generalized nonlinear models, for example).
