Closed-Form PMFs for PPCs

[ColtAllen]

While working on the upcoming BetaGeoBetaBinomModel, I found a closed-form solution in the research paper for the model's probability mass function (PMF). After doing some more digging I realized every CLV model has a closed-form expression.

Currently the distribution_new_customers method builds these PMFs from scratch via sampling. However, it should be more performant for PPCs (both prior and posterior) to use the closed-form PMF expressions.

[ColtAllen]

@ricardoV94 this sampling convention began with ShiftedBeteGeoModelIndividual. Is there a reason why the PMF in the paper wasn't used? See equation(3):

https://faculty.wharton.upenn.edu/wp-content/uploads/2012/04/Fader_hardie_jim_07.pdf

[ricardoV94]

@ricardoV94 this sampling convention began with ShiftedBeteGeoModelIndividual. Is there a reason why the PMF in the paper wasn't used? See equation(3):

https://faculty.wharton.upenn.edu/wp-content/uploads/2012/04/Fader_hardie_jim_07.pdf

Sampling from a PMF is not trivial? I'm not sure what you're suggesting

[ColtAllen]

Closing this because the closed-form PMF expressions are quite complex to code out and test, and it's uncertain how much faster this would be compared to the current sampling approach, the code for which is at least clean, easily adaptable & testable.