Add variances dispatch for distributions

[jessegrabowski]

Description

This PR adds a variance function that returns the (symbolic) variance of a distribution. I added it because I thought it would be nice to have for the AutoGuide stuff; we can use the mean and variance of priors to initialize the parameters.

This is a 90%+ LLM PR. I went through and tried to clean things up. I mostly copied the patterns from mean.py.

Related Issue

• Closes #
• Related to Add mean dispatch for pymc distributions #7530

Checklist

• Checked that the pre-commit linting/style checks pass
• Included tests that prove the fix is effective or that the new feature works
• Added necessary documentation (docstrings and/or example notebooks)
• If you are a pro: each commit corresponds to a relevant logical change

Type of change

• New feature / enhancement
• Bug fix
• Documentation
• Maintenance
• Other (please specify):

[codecov]

Codecov Report

❌ Patch coverage is 0.38610% with 258 lines in your changes missing coverage. Please review.
✅ Project coverage is 85.29%. Comparing base (5ff12c8) to head (1b0ee8f).
⚠️ Report is 11 commits behind head on main.

Files with missing lines	Patch %	Lines
pymc/distributions/moments/variances.py	0.00%	242 Missing ⚠️
pymc/distributions/moments/means.py	0.00%	11 Missing ⚠️
pymc/distributions/shape_utils.py	25.00%	3 Missing ⚠️
pymc/distributions/moments/__init__.py	0.00%	2 Missing ⚠️

❗ There is a different number of reports uploaded between BASE (5ff12c8) and HEAD (1b0ee8f). Click for more details.

HEAD has 1 upload less than BASE

Flag	BASE (5ff12c8)	HEAD (1b0ee8f)
	31	30

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #8084      +/-   ##
==========================================
- Coverage   90.80%   85.29%   -5.52%     
==========================================
  Files         121      124       +3     
  Lines       19443    19773     +330     
==========================================
- Hits        17656    16865     -791     
- Misses       1787     2908    +1121     

Files with missing lines	Coverage Δ
pymc/distributions/moments/__init__.py	0.00% <0.00%> (-100.00%)	⬇️
pymc/distributions/shape_utils.py	89.55% <25.00%> (-2.33%)	⬇️
pymc/distributions/moments/means.py	0.00% <0.00%> (-99.57%)	⬇️
pymc/distributions/moments/variances.py	0.00% <0.00%> (ø)

... and 18 files with indirect coverage changes

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[lucianopaz]

LGTM. My only suggestion is that we move the maybe_resize utility from means and variances to utils.py, but I'll approve anyway and you can move it or not.

[jessegrabowski]

I move the maybe_reize helper to distributions/shape_utils.py.

I also added variance for some missing distributions (CAR, Wishart, and KroneckerNormal).

Wishart was missing from means.py so I added it. The signature of KroneckerNormal was wrong in means, so I fixed it. And the formula for the mean of the rice distribution was wrong too, so I fixed that as well.

[lucianopaz]

Thanks for the fixes and improvements. I'm a bit unsure whether the variances should return inf or raise an error for Cauchy, Flat et al. I'm a bit more in favor of raising an error, but someone else should tie break.