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In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable X {\displaystyle X} equal to the number of failures needed to get r {\displaystyle r} successes in a sequence of independent Bernoulli trials. The probability p {\displaystyle 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 distributioncite-ref-johnson-1-0[1] or simply abbreviated as the BNB distribution. A shifted form of the distribution has been called the beta-Pascal distribution.cite-ref-johnson-1-1[1]

If parameters of the beta distribution are α α {\displaystyle \alpha } and β β {\displaystyle \beta } , and if

X ∣ ∣ p ∼ ∼ N B ( r , p ) , {\displaystyle X\mid p\sim \mathrm {NB} (r,p),}

where

p ∼ ∼ B ( α α , β β ) , {\displaystyle p\sim {\textrm {B}}(\alpha ,\beta ),}

then the marginal distribution of X {\displaystyle X} (i.e. the posterior predictive distribution) is a beta negative binomial distribution:

X ∼ ∼ B N B ( r , α α , β β ) . {\displaystyle X\sim \mathrm {BNB} (r,\alpha ,\beta ).}

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

• Definition and derivation

• PMF expressed with Gamma

• PMF expressed with the rising Pochammer symbol

• Properties

• Factorial Moments

• Non-identifiable

• Relation to other distributions

• Heavy tailed

• Beta geometric distribution

• Beta negative binomial as a Pólya urn model

• See also

• Notes

• References

• External links

Denoting f X | p ( k | q ) , f p ( q | α α , β β ) {\displaystyle 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 | α α , β β , r ) {\displaystyle f(k|\alpha ,\beta ,r)} of the BNB distribution by marginalization:

f ( k | α α , β β , r ) = ∫ ∫ 0 1 f X | p ( k | r , q ) ⋅ ⋅ f p ( q | α α , β β ) d q = ∫ ∫ 0 1 ( k + r − − 1 k ) ( 1 − − q ) k q r ⋅ ⋅ q α α − − 1 ( 1 − − q ) β β − − 1 B ( α α , β β ) d q = 1 B ( α α , β β ) ( k + r − − 1 k ) ∫ ∫ 0 1 q α α + r − − 1 ( 1 − − q ) β β + k − − 1 d q {\displaystyle {\begin{aligned}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\end{aligned}}}

Noting that the integral evaluates to:

∫ ∫ 0 1 q α α + r − − 1 ( 1 − − q ) β β + k − − 1 d q = Γ Γ ( α α + r ) Γ Γ ( β β + k ) Γ Γ ( α α + β β + k + r ) {\displaystyle \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 {\displaystyle r} is an integer, then the PMF can be written in terms of the beta function,:

f ( k | α α , β β , r ) = ( r + k − − 1 k ) B ( α α + r , β β + k ) B ( α α , β β ) {\displaystyle 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 | α α , β β , r ) = Γ Γ ( r + k ) k ! Γ Γ ( r ) B ( α α + r , β β + k ) B ( α α , β β ) {\displaystyle 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 | α α , β β , r ) = B ( r + k , α α + β β ) B ( r , α α ) Γ Γ ( k + β β ) k ! Γ Γ ( β β ) {\displaystyle f(k|\alpha ,\beta ,r)={\frac {\mathrm {B} (r+k,\alpha +\beta )}{\mathrm {B} (r,\alpha )}}{\frac {\Gamma (k+\beta )}{k!\;\Gamma (\beta )}}} .

Using the properties of the Beta function, the PMF with integer r {\displaystyle r} can be rewritten as:

f ( k | α α , β β , r ) = ( r + k − − 1 k ) Γ Γ ( α α + r ) Γ Γ ( β β + k ) Γ Γ ( α α + β β ) Γ Γ ( α α + r + β β + k ) Γ Γ ( α α ) Γ Γ ( β β ) {\displaystyle 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 | α α , β β , r ) = Γ Γ ( r + k ) k ! Γ Γ ( r ) Γ Γ ( α α + r ) Γ Γ ( β β + k ) Γ Γ ( α α + β β ) Γ Γ ( α α + r + β β + k ) Γ Γ ( α α ) Γ Γ ( β β ) {\displaystyle 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 )}}} .

The PMF is often also presented in terms of the Pochammer symbol for integer r {\displaystyle r}

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

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

E ⁡ ⁡ [ ( X ) k ] = Γ Γ ( r + k ) Γ Γ ( r ) Γ Γ ( β β + k ) Γ Γ ( β β ) Γ Γ ( α α − − k ) Γ Γ ( α α ) . {\displaystyle \operatorname {E} {\bigl [}(X)_{k}{\bigr ]}={\frac {\Gamma (r+k)}{\Gamma (r)}}{\frac {\Gamma (\beta +k)}{\Gamma (\beta )}}{\frac {\Gamma (\alpha -k)}{\Gamma (\alpha )}}.}

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

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

By Stirling's approximation to the beta function, it can be easily shown that for large k {\displaystyle k}

f ( k | α α , β β , r ) ∼ ∼ Γ Γ ( α α + r ) Γ Γ ( r ) B ( α α , β β ) k r − − 1 ( β β + k ) r + α α {\displaystyle 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 α α {\displaystyle \alpha } do not exist.

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

f ( k | α α , β β ) = B ( α α + 1 , β β + k ) B ( α α , β β ) {\displaystyle 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 β β = 1 {\displaystyle \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 ∼ ∼ B G ( α α , 1 ) {\displaystyle X\sim BG(\alpha ,1)} then X + 1 ∼ ∼ Y S ( α α ) {\displaystyle X+1\sim YS(\alpha )} .

In the case when the 3 parameters r , α α {\displaystyle r,\alpha } and β β {\displaystyle \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 α α {\displaystyle \alpha } red balls (the stopping color) and β β {\displaystyle \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 {\displaystyle r} red colored balls are drawn. The random variable X {\displaystyle X} of observed draws of blue balls are distributed according to a B N B ( r , α α , β β ) {\displaystyle \mathrm {BNB} (r,\alpha ,\beta )} . Note, at the end of the experiment, the urn always contains the fixed number r + α α {\displaystyle r+\alpha } of red balls while containing the random number X + β β {\displaystyle X+\beta } blue balls.

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

• Negative binomial distribution

• Dirichlet negative multinomial distribution

cite-note-johnson-11. ↑ Johnson et al. (1993)

• Johnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete Distributions, 2nd edition, Wiley ISBN 0-471-54897-9 (Section 6.2.3)

• Kemp, C.D.; Kemp, A.W. (1956) "Generalized hypergeometric distributions, Journal of the Royal Statistical Society, Series B, 18, 202–211

• Wang, Zhaoliang (2011) "One mixed negative binomial distribution with application", Journal of Statistical Planning and Inference, 141 (3), 1153-1160 doi:10.1016/j.jspi.2010.09.020

• Interactive graphic: Univariate Distribution Relationships