Beta negative binomial distribution
# Beta negative binomial distribution

In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable $X$ equal to the number of failures needed to get $r$ successes in a sequence of independent Bernoulli trials. The probability $p$ of success on each trial stays constant within any given experiment but varies across different experiments following a beta distribution. Thus the distribution is a compound probability distribution.

This distribution has also been called both the inverse Markov-Pólya distribution and the generalized Waring distribution or simply abbreviated as the BNB distribution. A shifted form of the distribution has been called the beta-Pascal distribution.

If parameters of the beta distribution are $\alpha$ and $\beta$, and if

    $X\mid p\sim \mathrm {NB} (r,p),$

where

    $p\sim {\textrm {B}}(\alpha ,\beta ),$

then the marginal distribution of $X$ is a beta negative binomial distribution:

    $X\sim \mathrm {BNB} (r,\alpha ,\beta ).$

In the above, $\mathrm {NB} (r,p)$ is the negative binomial distribution and ${\textrm {B}}(\alpha ,\beta )$ is the beta distribution.

## Definition and derivation

Denoting $f_{X|p}(k|q),f_{p}(q|\alpha ,\beta )$ the densities of the negative binomial and beta distributions respectively, we obtain the PMF $f(k|\alpha ,\beta ,r)$ of the BNB distribution by marginalization:

    $f(k|\alpha ,\beta ,r)=\int _{0}^{1}f_{X|p}(k|r,q)\cdot f_{p}(q|\alpha ,\beta )\mathrm {d} q=\int _{0}^{1}{\binom {k+r-1}{k}}(1-q)^{k}q^{r}\cdot {\frac {q^{\alpha -1}(1-q)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\mathrm {d} q={\frac {1}{\mathrm {B} (\alpha ,\beta )}}{\binom {k+r-1}{k}}\int _{0}^{1}q^{\alpha +r-1}(1-q)^{\beta +k-1}\mathrm {d} q$

Noting that the integral evaluates to:

    $\int _{0}^{1}q^{\alpha +r-1}(1-q)^{\beta +k-1}\mathrm {d} q={\frac {\Gamma (\alpha +r)\Gamma (\beta +k)}{\Gamma (\alpha +\beta +k+r)}}$

we can arrive at the following formulas by relatively simple manipulations.

If $r$ is an integer, then the PMF can be written in terms of the beta function,:

    $f(k|\alpha ,\beta ,r)={\binom {r+k-1}{k}}{\frac {\mathrm {B} (\alpha +r,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}$.

More generally, the PMF can be written

    $f(k|\alpha ,\beta ,r)={\frac {\Gamma (r+k)}{k!\;\Gamma (r)}}{\frac {\mathrm {B} (\alpha +r,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}$

or

    $f(k|\alpha ,\beta ,r)={\frac {\mathrm {B} (r+k,\alpha +\beta )}{\mathrm {B} (r,\alpha )}}{\frac {\Gamma (k+\beta )}{k!\;\Gamma (\beta )}}$.

### PMF expressed with Gamma

Using the properties of the Beta function, the PMF with integer $r$ can be rewritten as:

    $f(k|\alpha ,\beta ,r)={\binom {r+k-1}{k}}{\frac {\Gamma (\alpha +r)\Gamma (\beta +k)\Gamma (\alpha +\beta )}{\Gamma (\alpha +r+\beta +k)\Gamma (\alpha )\Gamma (\beta )}}$.

More generally, the PMF can be written as

    $f(k|\alpha ,\beta ,r)={\frac {\Gamma (r+k)}{k!\;\Gamma (r)}}{\frac {\Gamma (\alpha +r)\Gamma (\beta +k)\Gamma (\alpha +\beta )}{\Gamma (\alpha +r+\beta +k)\Gamma (\alpha )\Gamma (\beta )}}$.

### PMF expressed with the rising Pochammer symbol

The PMF is often also presented in terms of the Pochammer symbol for integer $r$

    $f(k|\alpha ,\beta ,r)={\frac {r^{(k)}\alpha ^{(r)}\beta ^{(k)}}{k!(\alpha +\beta )^{(r+k)}}}$

## Properties

### Factorial Moments

The k-th factorial moment of a beta negative binomial random variable X is defined for $k<\alpha$ and in this case is equal to

    $\operatorname {E} {\bigl(X)_{k}{\bigr ]}={\frac {\Gamma (r+k)}{\Gamma (r)}}{\frac {\Gamma (\beta +k)}{\Gamma (\beta )}}{\frac {\Gamma (\alpha -k)}{\Gamma (\alpha )}}.$

### Non-identifiable

The beta negative binomial is non-identifiable which can be seen easily by simply swapping $r$ and $\beta$ in the above density or characteristic function and noting that it is unchanged. Thus estimation demands that a constraint be placed on $r$, $\beta$ or both.

### Relation to other distributions

The beta negative binomial distribution contains the beta geometric distribution as a special case when either $r=1$ or $\beta =1$. It can therefore approximate the geometric distribution arbitrarily well. It also approximates the negative binomial distribution arbitrary well for large $\alpha$. It can therefore approximate the Poisson distribution arbitrarily well for large $\alpha$, $\beta$ and $r$.

### Heavy tailed

By Stirling's approximation to the beta function, it can be easily shown that for large $k$

    $f(k|\alpha ,\beta ,r)\sim {\frac {\Gamma (\alpha +r)}{\Gamma (r)\mathrm {B} (\alpha ,\beta )}}{\frac {k^{r-1}}{(\beta +k)^{r+\alpha }}}$

which implies that the beta negative binomial distribution is heavy tailed and that moments less than or equal to $\alpha$ do not exist.

## Beta geometric distribution

The beta geometric distribution is an important special case of the beta negative binomial distribution occurring for $r=1$. In this case the pmf simplifies to

    $f(k|\alpha ,\beta )={\frac {\mathrm {B} (\alpha +1,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}$.

This distribution is used in some Buy Till you Die (BTYD) models.

Further, when $\beta =1$ the beta geometric reduces to the Yule–Simon distribution. However, it is more common to define the Yule-Simon distribution in terms of a shifted version of the beta geometric. In particular, if $X\sim BG(\alpha ,1)$ then $X+1\sim YS(\alpha )$.

## Beta negative binomial as a Pólya urn model

In the case when the 3 parameters $r,\alpha$ and $\beta$ are positive integers, the Beta negative binomial can also be motivated by an urn model - or more specifically a basic Pólya urn model. Consider an urn initially containing $\alpha$ red balls (the stopping color) and $\beta$ blue balls. At each step of the model, a ball is drawn at random from the urn and replaced, along with one additional ball of the same color. The process is repeated over and over, until $r$ red colored balls are drawn. The random variable $X$ of observed draws of blue balls are distributed according to a $\mathrm {BNB} (r,\alpha ,\beta )$. Note, at the end of the experiment, the urn always contains the fixed number $r+\alpha$ of red balls while containing the random number $X+\beta$ blue balls.

By the non-identifiability property, $X$ can be equivalently generated with the urn initially containing $\alpha$ red balls (the stopping color) and $r$ blue balls and stopping when $\beta$ red balls are observed.

