added akaike and bayesian information criterion and documentation - #3700

Hi @astronomeralex thanks for contributing this!

Can I ask you to add some tests as well? It would be helpful to at least add one test for each tasks that test whether the correct value is returned for a given set of values.

It also would be helpful to test that the code properly handles if the code can correctly handle the different types of the variables passed to it (single floats or arrays ).

And it would appear that the build is currently failing on

 assert num_samples - num_params - 1 > 0

where it isn't sure if num_params is an array or float. This would be a good example of something that would need a test.

@crawfordsm right, yeah, num_params can be an array or a float. i'll fix that and add some tests today.

@crawfordsm -- after fighting tests and typos, i think this is ready to merge?

I wonder about how this could be added to astropy.modeling. For example, could we add this to the fit_info attached to some fitters? Would these even be useful for any of the existing fitters? I don't know the math here so I don't know the answers to these questions.

@astronomeralex - could you implement the fixes suggested by @hamogu, @adrn, and @embray?

@embray - I agree we should think about this, but this should not hold back this PR.

Is this the same as the aic and bic functions in statsmodels.tools.eval_measures (see here)?

If yes, I'm -1 to re-implement them in astropy.stats ... instead, point the astronomers to statsmodels from the astropy.stats docs here.

@astronomeralex Any time to update this PR. Failing that, is there anyone who can be assigned to it to ferry it through?

@astronomeralex just pinging you on this to see if you have some time to finalize this?

If @astronomeralex does not have time to finish this, and if you guys think this is still relevant whatsoever, I would like to be assigned to finish it up.

@mirca if you want to work on this next, that would be helpful.

Closing this after #4716 was merged.