Poisson Gamma Mixture = Negative Binomially Distributed?

Question

This paper introduces a model called "Beta-Geometric / NBD" which models "repeat-buying behavior in settings where customer “dropout” is unobserved: It assumes that customers buy at a steady rate (albeit in a stochastic manner) for a period of time, and then become inactive."

While I understand the "Beta-Geometric" aspects of the model (the number of purchases until a user becomes inactive, with Beta modeling heterogeneity across users), I don't quite get what NBD really refers to in this model.

NBD supposedly stands for Negative Binomial, but what's negative binomial about this model? I read online that it's related to having a Poisson-Gamma mixture, which this model has, but why? What's the connection between Negative Binomial and a Poisson Gamma mixture? If that's not the connection, why does the model name emphasize the term NBD?

I list the model assumptions below.

Related: Is it possible to understand pareto/nbd model conceptually?

Answer 1

There are various ways a negative binomial distribution can come about. One of them, as Robert Long comments, is as a Poisson distribution whose parameter is itself Gamma distributed. The Wikipedia page gives the derivation of this result. So this covers parts (i) and (ii) of your model.

This is an example of compound-distributions, which are often also called "mixtures" (e.g. a "Poisson-Gamma mixture" in the present case). This can be confusing, since a "mixture" has at least one related but distinct meaning in statistics.

Answer 2

The negative binomial distribution is the Poisson-gamma mixture. Specifically, it can be established that:

$$\text{NegBin} \bigg( t \bigg| n, \frac{1}{\theta+1} \bigg) = \int \limits_0^\infty \text{Pois}(t|\lambda) \ \text{Gamma}(\lambda|n, \theta) \ d \lambda.$$

(In this statement the parameter $\theta$ is the rate parameter of the gamma distribution.) This is a useful algebraic exercise to work through if you are new to these distributions.