Check the proper behavior of probability weighting normalization

[sami6mz]

As expressed in #42 and #43, we would like to check if normalization has a proper behavior.

"Normalization" refers to probability weighting normalization, as described in "Identifying causal mechanisms (primarily) based on inverse probability weighting", Huber (2014), DOI: 10.1002/jae.2341

We normalize the probabilities involved in score functions by dividing by their average on all the sample, respectively for treated and non-treated units. I describe that in my internship report page 22.

As far as I understood, it is a trick supposed to eliminate selection bias and make the estimation robust to extreme probability values.

A possible test would be to compare normalized and non-normalized estimators, or just to check the relative error like all other estimators (just add it to the tolerance dictionary).

As for now, normalization is implemented in med_dml and multiply_robust, which both give consistent results.
Normalization also is the very idea behind IPW estimator formulas.

[sami6mz]

By the way, @bthirion suggested to implement normalization into a function.
It is a bit tricky because it really depends on the estimator, but I guess it is doable.

[bthirion]

Sorry, my suggestion was actually a bit different (sorry if that was unclear): when you have 10-15 lines of code running a computation (any computation), you should cut that into an ancillary function. The point is that you can test that small function independently + it makes code more readable.

[judithabk6]

this has been implemented, and included in the tests as normalize is the default behavior, so not sure extra exploration or tests are needed as tests are already too long (#52). Probably should be included in a general discussion on tests