# Poisson Gamma Mixture = Negative Binomially Distributed?

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.

• The Negative Binomial is equivalent to a Poisson(λ) distribution, where λ is a random variable, distributed as a gamma distribution with shape = r and scale θ – Robert Long Jul 27 '20 at 4:25

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