Fix parameterization in _distribution_new_customers

[wd60622]

Address #328

Wondering why there is discrepancy in the generative process in the top of the test class and what is required here. Also, this test doesn't confirm the validity

TODO:

• Add test to confirm @ColtAllen 's claim. Compare the rate to the scale of the observed data
• Is it good to add an example and a plot of this to one of the notebooks? visual confirmation of above

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📚 Documentation preview 📚: https://pymc-marketing--430.org.readthedocs.build/en/430/

[codecov]

Codecov Report

Merging #430 (04d2462) into main (e9a854d) will not change coverage.
The diff coverage is 100.00%.

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pymc_marketing/clv/models/beta_geo.py	99.09% <100.00%> (ø)

[wd60622]

I still think the parameterization was correct before. For instance, the distribution for new population_purchase_rate is in the same scale of naive empirical estimates (obviously better as dropout is possible).
On the otherhand, the 1 / alpha parameterization is way larger.
I had that population_purchase_rate is the rate / time unit so the previous parametrization makes sense to me

Would it make sense to assert

1. population_purchase_rate estimate <= (frequency / T).max()
2. some relation between empirical and rate estimates

[ColtAllen]

Frequency/recency would make for a better emperical comparison, as recency denotes the time period of the customer's last purchase. However, I've been thinking of this in the reciprocal (recency/frequency) which would be the average number of days between purchases.

The best way to test this in practice (if not for this PR) would be to do a posterior predictive check with these purchase and dropout rate distributions as parametrized:

However, PPCs aren't supported in BetaGeoModel. Doing so would require resolving this issue:

#128

[wd60622]

Sure, that is better approximation, but I would argue the previous parameterization makes more sense with even that value.

For instance, the scale of the previous parameterization goes from 0 to 4 purchases / unit time while alive whereas the suggested parameterization has that goes from 0 to 30+ / unit time. Both the empirical frequency / recency and frequency / T distributions are in that range.

If we had the model where dropout doesn't happen and keep poisson likelihood (even without the gamma population prior), one of those would be our MLEs (the gamma would regularize) so the range of the previous parameterization makes most sense to me.

Pardon any typos in the graphs 🥲

This is all from the test generated data which has those same parameter assumptions, but those ranges just don't add up for me

[ColtAllen]

Thinking more on this, I've realized you're right - this model is a poisson process for frequency over time, and flipping the rate parameter won't yield time per frequency.

The parameter can remain as previously defined; no need for changes. I do think time between purchases is more interpretable to the user though, so I'll make a note to take the reciprocal of this function's output for plotting in a future PR.

One last nit-pick - the tests in beta geo use arbitrary parameters rather than generating them from a test dataset, and r=4 is unrealistic. In every dataset I've worked with that parameter hovers around 0.5.

[wd60622]

All good. I was wrapping my head around it too. Between reusing greek letters and gamma parameterizations, its hard to keep it straight.
Thanks for taking a look!

[wd60622]

One last nit-pick - the tests in beta geo use arbitrary parameters rather than generating them from a test dataset, and r=4 is unrealistic. In every dataset I've worked with that parameter hovers around 0.5.

Might be good for a new issue?