Update PMF example

[ckrapu]

Update Probabilistic Matrix Factorization example

Address #743 and #58, updating this notebook to run cleanly and also performing some light edits for clarity and correctness.

• Notebook follows style guide https://docs.pymc.io/en/latest/contributing/jupyter_style.html
• PR description contains a link to the relevant issue:
  • a tracker one for existing notebooks (tracker issues have the "tracker id" label)
  • or a proposal one for new notebooks
• Check the notebook is not excluded from any pre-commit check: https://github.com/pymc-devs/pymc-examples/blob/main/.pre-commit-config.yaml

Helpful links

• https://github.com/pymc-devs/pymc-examples/blob/main/CONTRIBUTING.md

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[ckrapu]

The only substantial change in the model for this notebook was a replacement of the MvNormal RVs which had a diagonal covariance matrix with independent draws from Normal instead.

One odd thing that I found was that the MAP optimization finished in <1 second on the old notebook, but took 40 seconds this time. Also, the performance difference between the mean-of-means baseline and the MAP estimate was not reproducible, but the answer we get now (MAP is closer to baseline performance instead of much worse) makes more sense to me. I also added a few bullet points to explain the dims of the various matrices.

Running MCMC for the main model takes roughly 2x as long now as well.

[ckrapu]

Any feedback on these edits yet?

[aloctavodia]

Thanks @ckrapu!