I was recently trying to understand [generalized linear models](https://en.wikipedia.org/wiki/Generalized_linear_model) (GLMs) and after investing quite a few days, it still hasn't dawned on me what the fundamental benefit of the framework is. Normally, I am used to results like guarantees of convergence, limits for error etc, but all that seems to be missing here.


There is a common framework with underlying distribution, regressors/predictors linear in the coefficients, link functions and finally MLE but it seems to be branching off very quickly into the various subclasses, which each need a separate algebraical and numerical treatment.


So can anyone point me towards what is "general" about the GLMs and what is the benefit of that?



Generalized linear models: What's the benefit of the underlying theory?
What are the benefits of a unified framework? You are right that we are rapidly going into some much used special cases line logistic regression or Poisson regression, but there is still benefit in having a common framework.


* Technology transfer from the *general linear model* (**not** generalized!), that is with gaussian errors and identity link. A lot of what one has learned from there can be directly used with glm's, especially all the modeling trick constructing the model matrix $X$.
* A common estimation algorithm (IRLS, iteratively reweighted least squares, see <https://stats.stackexchange.com/questions/236676/can-you-give-a-simple-intuitive-explanation-of-irls-method-to-find-the-mle-of-a/237384#237384>), leads to a common implementation framework. This is maybe not a mathematical advantage, but software implementation and modeling advantage. It is also a teaching advantage! This program is not a 100% success, as for some important glm's this algorithm do not work very well, as witnessed by the paper [logbin: An R Package for Relative Risk Regression Using the Log-Binomial Model](https://www.jstatsoft.org/article/view/v086i09).
* Many common concepts applied to all or most of the special cases, like link function, offsets, variance function, mean function, ... , quasi-likelihood
* One area with little advantage of the common framework is residual analysis, which really need to be studied for each case separately. See for instance [family of glm represents the distribution of the response variable or residuals](https://stats.stackexchange.com/questions/374452/family-of-glm-represents-the-distribution-of-the-response-variable-or-residuals/374461#374461). But see here for an approach [using simulated residuals](https://stats.stackexchange.com/questions/70558/diagnostic-plots-for-count-regression/248035#248035).
* The Nelder & Wederburn paper introding the glm is [here at JSTOR](https://www.jstor.org/stable/pdf/2344614.pdf) and the original motivations can be found there.


A lot of information can be found at Cross Validated, see as a start <https://stats.stackexchange.com/questions/104399/why-do-we-use-glm>


