Revamp our negative binomial explanation

[canyon289]

Description of your problem

Our explanation of negative binomial parametrisation is either confusing, or wrong, I don't know which.

More specifically the explanation of the mu parameter itself is confusing, I personally dont understand how poisson is being parameterized twice, both by gamma samples controlled by alpha, and mu

Stans documentation does a much better job explaining the effect of each parameter, perhaps we can incorporate some of their wording
https://mc-stan.org/docs/2_20/functions-reference/nbalt.html

[canyon289]

From @junpenglao

It's because pymc is explaining the parameterzation with NegBino being Gamma Poisson mixture https://en.wikipedia.org/wiki/Negative_binomial_distribution#Gamma%E2%80%93Poisson_mixture, and Stan is explaining NegBino being overdispersed Poisson (https://en.wikipedia.org/wiki/Negative_binomial_distribution#Poisson_distribution and https://en.wikipedia.org/wiki/Negative_binomial_distribution#Overdispersed_Poisson)

[samO2I1]

@canyon289 can I work on this issue?

[canyon289]

Please do! Let us know how we can help

[samO2I1]

@canyon289 can you please share the source code link?

[canyon289]

Here it is https://github.com/pymc-devs/pymc/blob/main/pymc/distributions/discrete.py#L665

[samO2I1]

@canyon289 I am not able to understand that at which points Poisson is parameterized twice?

[canyon289]

The documentation somewhat confusingly describes

• The poisson distribution is parametrized by the gamma distribution
• The poisson distribution (somehow is also) parameterized by mu
• The gamma distribution is parameterized by alpha

I've provided a directly link here. To be honest I don't know what the correct explanation is so I won't be able to provide much guidance there.
https://docs.pymc.io/en/v3/api/distributions/discrete.html#pymc3.distributions.discrete.NegativeBinomial

[herve-91]

Hello

In the context, considered in the documentation, where the negative binomial is introduced as a poisson random variable whose rate is gamma distributed, I believe both and should be presented as parameters of the gamma distribution:

• is the shape of the gamma distribution
• is the mean of the gamma distribution.

Indeed, if we consider a poisson distribution whose rate is gamma distributed:

where

• is a poisson distribution with rate

• is a gamma distribution with shape parameter and rate parameter

We have:

from which we can recover the formula of the documentation if we replace by where is the mean of the gamma distribution :

Regards,

[canyon289]

This is great and much less confusing. Would you like to add some version of it the docs? Thank you for investigating!

[canyon289]

Thanks @herve-91