Improve factor analysis notebook

[erik-werner]

Fixes #587

Updating factor analysis notebook to

• Fix some small errors in the mathematical derivations and related code

• Spell out some steps in the derivations that were not obvious (to me at least)

• Replace the least-squares derivation of $F$ by a Bayesian derivation

• Add a section at the end that compares the factors we have identified to the true ones for the synthetic data we have used

• Do some minor edits to the text, in particular to fit the text to the updated sampling results

• [x] Notebook follows style guide https://docs.pymc.io/en/latest/contributing/jupyter_style.html

• [x] 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

• [x] Check the notebook is not excluded from any pre-commit check: https://github.com/pymc-devs/pymc-examples/blob/main/.pre-commit-config.yaml

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:books: Documentation preview :books:: https://pymc-examples--612.org.readthedocs.build/en/612/

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[twiecki]

Thanks!