Is `TweedieRegressor` a completely general GLM solution?

Question

I consider sklearn's TweedieRegressor a general solution for all types of regression, as it is a GLM model. If I understand well, that means any regression type can be obtained by appropriately parametrizing TweedieRegressor.

Logistic regression is a special case of GLM. I guess, to have logistic regression there should be a logit value (or something like that) for the link parameter, but there is no such possibility.

Can TweedieRegressor be parametrized to do logistic regression?
If yes, how? If not, why not?

Answer 1

No. A GLM is characterized by its link function and its target distribution. TweedieRegressor assumes a Tweedie distribution for the latter, which do not include the Bernoulli distribution needed for logistic regression.

Answer 2

Logistic regression corresponds to a Binomial distribution, a member of the exponential family, so in that sense it is nested within the Exponential Dispersion class of models. The $\mathrm{Tweedie}(\mu, \sigma^2)$ family specifically is also contained within $\mathrm{ED}$, but imposes the mean-variance relationship

\begin{align*} \mu &= \mathbf{E}[Y]\\ \operatorname{var}(Y) &= \sigma^2 \mu^p. \end{align*}

This cannot work for logistic regression because the Binomial needs $$\operatorname{var}(Y) = np(1-p) = n\mathbf{E}[Y](1-\mathbf{E}[Y]).$$