Reason for exponential as nonlinearity in GLM

Hi all! Why is the exp function used as non-linear element in Poisson GLM? I found some answers related to this questions here, but I can’t yet build an intuition on that:

Could we use another function as well? Which are the criteria for selecting the nonlinear function?


Apparently failed to hit “Reply” on this about a month ago, so in case you’re interested:

It’s been most of a decade since I studied point processes, so take this with a grain of salt, but IIRC the main reason you have a link function with some given properties (judging by exp, probably positivity and monotonicity) is to ensure that the optimization surface is convex, which means it’s amenable to finding a global optimum via gradient descent.

If you want to know “Why exponentials instead of another class of functions”, I would say mathematicians like things that are easy to deal with. Exponential functions tend to be nice in terms of their derivatives and integrals, and allow you to use additive terms in the exponent as multiplicative terms in the model.

If the question is “Why base e instead of another base?” the answer is probably that it means not having to throw in a scaling factor every time you differentiate/integrate. d/dx e^f(x) == (d/dx f(x)) * e^f(x)

Not sure if this answers your question, really.

It helps a lot, thanks!