Implement Ordered distribution factory

[ricardoV94]

Description

For univariate IID, adding a transform=ordered is equivalent to sorting the raw draws (forward pass). The logp is proportional to the density of the original draws + ordered transform jacobian. We would just need to figure out the normalization constant so it defines a proper multivariate variable that integrates to 1.

With this users would have a generative graph for ordered variables, that they can also do prior_predictive sampling from.
Also default initvals would work out of the box. Right now users always need to provide them.

This would also pretty much also obliviate the need for default_transform and transform that we added in #5674 and simplify the API.

The normalization constant is probably just size!: https://en.wikipedia.org/wiki/Order_statistic#The_joint_distribution_of_the_order_statistics_of_an_absolutely_continuous_distribution

The API could look something like:

with pm.Model() as m:
  x = pm.Ordered("x", pm.Normal.dist(), shape=(3,))

[michaelosthege]

Just stressing (citing you) that

Transforms have no role in forward sampling methods (prior/posterior predictive)

from #7040 (comment) because we just got startled by this (again).

[ricardoV94]

Just stressing (citing you) that

Transforms have no role in forward sampling methods (prior/posterior predictive)

from #7040 (comment) because we just got startled by this (again).

This is now emphasized in https://www.pymc.io/projects/docs/en/latest/api/distributions/transforms.html

With the distinction between transform and default_transform we can also start emitting warnings in prior/posterior predictive sampling like we do with Potentials, that non default transforms will be ignored

[AlexAndorra]

Also default initvals would work out of the box. Right now users always need to provide them.

What do you mean by that @ricardoV94 ? I have definitely had cases where it worked without specifying initivals, IIRC

[ricardoV94]

Also default initvals would work out of the box. Right now users always need to provide them.

What do you mean by that @ricardoV94 ? I have definitely had cases where it worked without specifying initivals, IIRC

Then you were lucky, or had a non-iid prior like mu=[-1, 0, 1]

[AlexAndorra]

Aaaah right, makes more sense now -- I indeed always order the mus. Thanks Ricardo ;)

[ricardoV94]

Note that my suggestion above only works for iid components. It wouldn't accept stuff with different mus for example