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Just wanted to see if I understood the meaning of Generalized Linear Models:

In the case of Standard ( "Non-generalized") Linear models, a dependent variable y is a linear function of a dependent variable x. In a Generalized Linear Model (GLM), a dependent variable y is linear in some function L of the variable x, i.e., y is linear in L(x)? I am thinking of the example case of Logistic Regression as an example of GLM where y depends linearly on the log-odds : Log(p/(1-p))? Do we have multilinear cases? Is this accurate, and if so, complete, essentially?

Thanks.

EDIT: Also, is there some result tellings us given data pairs ##\{(x_i,y_i) \}; y_i \~ x_i##, i.e., y depends on x, when we can find a function L so that y is linear in L(x)?