Beta negative binomial distribution explained

In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable

equal to the number of failures needed to get

successes in a sequence of independent Bernoulli trials. The probability

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^[1] or simply abbreviated as the BNB distribution. A shifted form of the distribution has been called the beta-Pascal distribution.^[1]

If parameters of the beta distribution are

\alpha

and

\beta

, and if

X\midp\simNB(r,p),

where

p\simrm{B}(\alpha,\beta),

then the marginal distribution of

(i.e. the posterior predictive distribution) is a beta negative binomial distribution:

X\simBNB(r,\alpha,\beta).

In the above,

NB(r,p)

is the negative binomial distribution and

rm{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)=

	1
\int
	0

f_X|p(k|r,q) ⋅ f_p(q|\alpha,\beta)dq=

	1
\int
	0

\binom{k+r-1}{k}(1-q)^kq^r ⋅

	q^\alpha-1(1-q)^\beta-1
	\Beta(\alpha,\beta)

dq=

	1
	\Beta(\alpha,\beta)

\binom{k+r-1}{k}

	1
\int
	0

q^\alpha+r-1(1-q)^\beta+k-1dq

Noting that the integral evaluates to:

	1
\int
	0

q^\alpha+r-1(1-q)^\beta+k-1dq=

	\Gamma(\alpha+r)\Gamma(\beta+k)
	\Gamma(\alpha+\beta+k+r)

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

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	\Beta(\alpha+r,\beta+k)
	\Beta(\alpha,\beta)

.More generally, the PMF can be written

f(k\|\alpha,\beta,r)=	\Gamma(r+k)
	k! \Gamma(r)

	\Beta(\alpha+r,\beta+k)
	\Beta(\alpha,\beta)

f(k\|\alpha,\beta,r)=	\Beta(r+k,\alpha+\beta)
	\Beta(r,\alpha)

	\Gamma(k+\beta)
	k! \Gamma(\beta)

PMF expressed with Gamma

Using the properties of the Beta function, the PMF with integer

can be rewritten as:

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

	\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

f(k\|\alpha,\beta,r)=	r^(k)\alpha^(r)\beta^(k)
	k!(\alpha+\beta)^(r+k)

Properties

Factorial Moments

The -th factorial moment of a beta negative binomial random variable is defined for

k<\alpha

and in this case is equal to

\operatorname{E}l[(X)_kr]=

	\Gamma(r+k)
	\Gamma(r)

	\Gamma(\beta+k)
	\Gamma(\beta)

	\Gamma(\alpha-k)
	\Gamma(\alpha)

Non-identifiable

The beta negative binomial is non-identifiable which can be seen easily by simply swapping

and

\beta

in the above density or characteristic function and noting that it is unchanged. Thus estimation demands that a constraint be placed on

\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

\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

Heavy tailed

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

f(k|\alpha,\beta,r)\sim

	\Gamma(\alpha+r)
	\Gamma(r)\Beta(\alpha,\beta)

	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)=	B(\alpha+1,\beta+k)
	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\simBG(\alpha,1)

then

X+1\simYS(\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

red colored balls are drawn. The random variable

of observed draws of blue balls are distributed according to a

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,

can be equivalently generated with the urn initially containing

\alpha

red balls (the stopping color) and

blue balls and stopping when

\beta

red balls are observed.

References

Johnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete Distributions, 2nd edition, Wiley (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

External links

Interactive graphic: Univariate Distribution Relationships

Notes and References

Johnson et al. (1993)

Beta negative binomial distribution explained

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

References

External links

Notes and References