Remove link argument from TweedieRegressor

Proposal

Remove the argument link from TweedieRegressor and always use a log-link.

Rationale

While PoissonRegressor and GammaRegressor have built-in log-link (prediction = exp(…)), the TweedieRegressor has an argument link : {'auto', 'identity', 'log'}, default='auto'.

The problem is that Tweedie should always predict positive vallues, but with link=identity`’ this goes wrong, for example

import numpy as np
from sklearn.linear_model import TweedieRegressor


y = np.array([3, 2, 1, 0, 0])
X = np.arange(5)[:, None]
reg = TweedieRegressor(power=1, link='identity')  # power=1 optimizes Poisson deviance
reg.fit(X, y)
reg.predict(X)

Out[7]: array([ 1.95131702,  0.45263687, -1.04604328, -2.54472343, -4.04340359])

The only instance, where negatives predictions are ok, is power=0 which gives squared error. But if someone want’t to fit squared error, she or he should better use Ridge and the likes.

[lorentzenchr]

It means that this issue is open for discussion and needs a decision by the core devs. A PR/implementation only makes sense after such a decision.

[jpbrodsky]

My use case for an identity link: (Which I also mentioned at https://github.com/scikit-learn/scikit-learn/pull/22548#discussion_r869765737)

I have a problem where I know the expectation y’ follows the linear model y’ = w x. My measurements, y, have poisson errors.

(The specific problem involves analysis of radiation measurements. The expectation is linear with the amount of source; the measurements are poisson distributed).

Using a log link function is just not the right description of my problem. Yes, the whole thing breaks down when evaluating negative values of w, but it seems much better to offer a constraint to avoid ever evaluating negative values of w rather than exclude the situations where you have an actual linear relationship with poisson measurements.