How to derive Negative Binomial process from Poisson process? I am trying to understand NB process, and how it can be derived from Poisson process.
Zhou & Carin states that:
"By placing a gamma prior with shape $r$ and
scale $\frac{p}{1−p}$ on $λ$ as $m \sim Pois(λ)$, $λ \sim Gamma(r, \frac{p}{1−p})$ and marginalizing out $λ$, an NB distribution $m \sim NB(r,p)$ is obtained, with PMF
$f_{M} (m|r,p) = \frac{Γ(m|r,p)}{m!Γ(r)}(1 - p)^rp^m,   m \in  Z_{+}$
Now my question is:
Does that mean that to construct a NB process model, I should specify:
$m \sim Pois(λ)$        (1)
or
$f_{M} (m|r,p) = \frac{Γ(m|r,p)}{m!Γ(r)}(1 - p)^rp^m,   m \in  Z_{+}$       (2)
or
both (1) and (2)?
Thanks in advance.
 A: That part of the article has nothing to do with Poisson process or negative binomial process. What they are doing is deriving the negative binomial distribution from a gamma prior for $\lambda$ like so.
$$\begin{split}f(m,\lambda|r,p)&=f(\lambda|r,p)f(m|\lambda)\\
&=\frac{\left(\frac{p}{1-p}\right)^r}{\Gamma(r)}\lambda^{r-1}e^{-\frac p{1-p}\lambda}\frac{e^{-\lambda}}{\lambda ^m}m!\end{split}$$
Now marginalize out $\lambda$:
$$\begin{split}\int_0^\infty e^{-\frac\lambda{1-p}}\lambda^{m+r-1}d\lambda&=(1-p)^{m+r-1}\int_0^\infty e^{-\frac\lambda{1-p}}\left(\frac\lambda{1-p}\right)^{m+r-1}d\lambda\\
&=(1-p)^{m+r}\int_0^\infty e^{-u}u^{m+r-1}du\\
&=(1-p)^{m+r}\Gamma(m+r)\end{split}$$
so that $$\begin{split}f(m|r,p)&=\frac{\Gamma(m+r)}{\Gamma(r)m!}p^r(1-p)^m\end{split}$$
This is the pdf of a $\text{Negative Binomial}(r, p)$ distribution.
But if you were interested in a negative binomial process, its relation to the poisson process is as follows. Suppose $N_t$ is the number of arrivals in a period of time $t$. We know that $Negative Binomial(r, p)\approx Poisson(\lambda)$ if $r(1-p)=\lambda$, and further that $r$ is large and $p$ is close to $1$. So if a Poisson process has rate $\lambda$, then the negative binomial process that models arrivals in an interval of length $t$ as $Negative Binomial(rt, p)$ will be approximately the same as the poisson process subject to said condition on $r$ and $p$.
