So I'm a 3rd year undergraduate doing my thesisin football score models right now. In my thesis I want to include a proof of what the link function for the Poisson distribution is and why it relates the mean to our linear predictors. I'm almost there, but there is one part that most literature seems to gloss over.
So we have our linear predictors η =β0 + β1_xi1+⋯+βp_xip and our natural parameter θ. Now I know why if η = θ then our link function is canonical, but my question is how do we prove that we have η = θ in the case of the Poisson distribution? Most literature just state that the canonical link sets this equality, or we can assume this equality for distributions that are members of the exp. family but don't actually prove why. Can anyone help?