Added Probabilistic PCA example

[ParthsarthiSingh-glang]

Summary

Adds a comprehensive Probabilistic PCA example to the case studies section.

Closes #250

Contents

• Mathematical background on PPCA
• Full PyMC implementation with priors
• Comparison with classical PCA (sklearn)
• Convergence diagnostics and trace plots
• Discussion of rotational identifiability

Testing

• Notebook runs successfully end-to-end
• All cells execute without errors
• Visualizations render correctly

Notes

• ESS warnings are expected due to rotational non-identifiability
• Tested on Windows with Python 3.13

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[ParthsarthiSingh-glang]

@twiecki please review it sir so that i can work it on further if issues exists.
Thanks.

[twiecki]

The convergence doesn't look the best, did you investigate?

[ParthsarthiSingh-glang]

@twiecki Thanks for taking a look!

You're right - the convergence warnings are there because of the rotational symmetry in PPCA. Since you can rotate the latent space and get the same answer, the sampler has trouble settling down on one specific solution.

But the model is working correctly - the reconstructions look good and match what sklearn's PCA produces. It's just that the diagnostics don't look great because of this rotation issue.

Would it help if :

• Increase the sampling to get better numbers?
• Add a note explaining why this happens with PPCA?

I can adjust on whatever you tell .

[twiecki]

Yes, I'd add a note explaining that.