Invalid values being used for testing Beta distribution

[kc611]

Description of your problem

This was pointed out in #4887 that the testing for Beta distribution used values that might lead to invalid combination of arguments in the following test:

https://github.com/pymc-devs/pymc3/blob/53e572caa3144f77951e233f6bcca72b63084bd8/pymc3/tests/test_distributions.py#L1164-L1169

The values for mu and sigma are picked randomly from their respective Domain. However in this particular case we only have certain value pairs which are valid. In the current v4 testing it simply returns NaN values for invalid cases. With #4887 we will require a new mechanism/value Domains to be placed over here.

[ricardoV94]

@lucianopaz Suggested we should drop the sigma parameterization altogether, and use mu + nu instead, with nu being what we call kappa in here: https://github.com/pymc-devs/pymc3/blob/53e572caa3144f77951e233f6bcca72b63084bd8/pymc3/distributions/continuous.py#L1224

Wikipedia calls this mean + sample size: https://en.wikipedia.org/wiki/Beta_distribution#Mean_and_sample_size
This avoids the hard constraints of sigma that are making the tests fail (and it's also easier to work with).

[lucianopaz]

Just to add on this, from what I found this is called the mean and sample size parametrization. The advantage that it has is that the mu and nu bounds are independent from each other: mu must be in the (0, 1) interval and nu must only be positive.