Deprecate sample_posterior_predictive_w

[zaxtax]

This PR is contingent on arviz-devs/arviz#2147 being merged. But I think arviz.stats.weight_predictions is a better API for model averaging than pm.sample_posterior_predictive_w

The main thing that's need in addition is to update the notebook to use the arviz function, and another PR to update examples that used to use pm.sample_posterior_predictive_w

Checklist

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Major / Breaking Changes

• Removes sample_posterior_predictive_w which had not been refactored for v4. Also see Add weight_predictions function arviz-devs/arviz#2147 and Deprecate sample_posterior_predictive_w #6254.

Docs / Maintenance

• No Docs need to change

[codecov]

Codecov Report

Merging #6254 (9f39081) into main (bcffce2) will increase coverage by 3.56%.
The diff coverage is n/a.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #6254      +/-   ##
==========================================
+ Coverage   90.57%   94.14%   +3.56%     
==========================================
  Files         100      100              
  Lines       21235    21235              
==========================================
+ Hits        19234    19992     +758     
+ Misses       2001     1243     -758     

Impacted Files	Coverage Δ
pymc/backends/base.py	87.65% <0.00%> (-0.83%)	⬇️
pymc/parallel_sampling.py	87.03% <0.00%> (+0.34%)	⬆️
pymc/distributions/distribution.py	95.00% <0.00%> (+0.45%)	⬆️
pymc/aesaraf.py	93.54% <0.00%> (+2.32%)	⬆️
pymc/tests/distributions/util.py	90.86% <0.00%> (+2.46%)	⬆️
pymc/distributions/logprob.py	97.26% <0.00%> (+3.42%)	⬆️
pymc/tuning/starting.py	92.50% <0.00%> (+12.50%)	⬆️
pymc/tuning/scaling.py	62.74% <0.00%> (+25.49%)	⬆️
pymc/distributions/dist_math.py	87.42% <0.00%> (+29.71%)	⬆️
pymc/distributions/transforms.py	100.00% <0.00%> (+31.42%)	⬆️
... and 5 more

[michaelosthege]

Given that the function raised a NotImplementedError beforehand, I think we can merge without waiting on the ArviZ release.

What do you think, @aloctavodia ?

[aloctavodia]

Sounds good to me.