Beta binomial distribution

The beta binomial distribution is a special probability distribution in stochastics . It is one of the discrete probability distributions and is univariate . It can be seen as a kind of generalization of the binomial distribution , since it specifies the probability of successes given a given probability of a single success, while in the beta binomial distribution the probability of success is only imprecisely known and is described by a beta distribution B (a, b) becomes. It is therefore a mixed distribution . ${\ displaystyle x}$ ${\ displaystyle n}$

The beta binomial distribution has three parameters: n , a , b

definition

The probability function of the beta binomial distribution for different parameters

The distribution function of the beta binomial distribution for different parameters

A random variable has a beta binomial distribution with the parameters , and , in characters , if they are for all out of the carrier the probability function ${\ displaystyle X}$ ${\ displaystyle n \ in \ mathbb {N} _ {0}}$ ${\ displaystyle a> 0}$ ${\ displaystyle b> 0}$ ${\ displaystyle X \ sim BeB (n, a, b)}$ ${\ displaystyle x}$ ${\ displaystyle \ {0,1, \ ldots, n \}}$

{\ displaystyle P (X = x) = {n \ choose x} {\ frac {\ mathrm {B} (a + x, b + nx)} {\ mathrm {B} (a, b)}}}

has, where is the beta function . ${\ displaystyle \ mathrm {B}}$

construction

If the probability function is the binomial distribution and the density is the beta distribution, then the probability function of the mixed distribution is calculated as ${\ displaystyle f (x | p, n)}$ ${\ displaystyle b (p | a, b)}$

{\ displaystyle P (X = x) = M (x | n, a, b) = \ int _ {0} ^ {1} f (x | p, n) b (p | a, b) dp}

.

The integral corresponds exactly to the above probability function.

Alternative representation

Alternatively, the probability function can also be represented as

{\ displaystyle P (X = x) = C {n \ choose x} \ Gamma (a + x) \ Gamma (b + nx).}

The constant C is a normalization constant and is calculated as follows:

{\ displaystyle C = {\ frac {\ Gamma (a + b)} {\ Gamma (a) \ Gamma (b) \ Gamma (a + b + n)}}}

Here is the gamma function . ${\ displaystyle \ Gamma}$

properties

Expected value

The expected value depends on all three parameters:

{\ displaystyle E (X) = n {\ frac {a} {a + b}},}

Variance

The variance is:

{\ displaystyle Var (X) = n {\ frac {ab} {(a + b) ^ {2}}} {\ frac {a + b + n} {a + b + 1}}.}

Crookedness

The skew is indicated with

{\ displaystyle \ operatorname {v} (X) = (a + b + 2n) {\ frac {ba} {a + b + 2}} {\ sqrt {\ frac {1 + a + b} {nab (n + a + b)}}}}

Probability generating function

The probability generating function of the beta binomial distribution is

{\ displaystyle m_ {X} \ left (t \ right) = {_ {2} F_ {1}} \ left (-n, a; a + b; 1-t \ right) \!}

.

Here is the Gaussian hypergeometric function . ${\ displaystyle _ {2} F_ {1}}$

Characteristic function

The characteristic function follows through substitution :

{\ displaystyle \ varphi _ {X} \ left (t \ right) = {_ {2} F_ {1}} \ left (-n, a; a + b; 1-e ^ {it} \ right) \ !}

.

Moment generating function

This is the moment generating function

{\ displaystyle M_ {X} \ left (t \ right) = {_ {2} F_ {1}} \ left (-n, a; a + b; 1-e ^ {t} \ right) \!}

.

Special cases

If and , then it is a discrete uniform distribution with , since the carrier contains values. ${\ displaystyle a = 1}$ ${\ displaystyle b = 1}$ ${\ displaystyle P (X = x) = {\ tfrac {1} {n + 1}}}$ ${\ displaystyle n + 1}$

Areas of application

The beta binomial distribution is typically used in cases in which a binomial distribution would normally be used, but cannot assume that all individual events have the same probability of occurring, but that these probabilities are more or less bell-shaped around a value.

For example, if you want to know how many light bulbs will fail within the next 12 months, but assume that the probability of a light bulb failing varies between different delivery boxes, then a beta binomial distribution is appropriate.

Empirically one can assume to have to do with a beta binomial distribution, although one would rather think of a binomial model if the data spreads more than intended by the binomial distribution.

example

Model in Bayesian statistics

An urn contains an unknown number of balls, of which we know from other samples that the proportion of red balls is described by a beta distribution. ${\ displaystyle B (a, b)}$

There are n are pulled times balls (with replacement). The probability that a red ball will be drawn x times is in the beta binomial distribution . ${\ displaystyle BeB (n, a, b)}$

Numerical example

Based on a complete ignorance of the a priori distribution, which is described with a (alternatives are, for example ), a "preliminary study" is organized with a drawing (with repetition) of 15 balls. One of these balls is red. Thus the a posteriori distribution is described with the . ${\ displaystyle Beta (1,1)}$ ${\ displaystyle Beta ({\ tfrac {1} {2}}, {\ tfrac {1} {2}})}$ ${\ displaystyle Beta (1 + 1.1 + 14) = Beta (2.15)}$

The actual "study" provides for a drawing of 40 balls. What is asked is the probability that a red ball will be drawn exactly twice.

Since in this second drawing the probability is that one , it can be calculated as follows: ${\ displaystyle P (X = x)}$ ${\ displaystyle BeB (40,2,15)}$

{\ displaystyle P (X = 2, n = 40, a = 2, b = 15) = C {40 \ choose 2} \ Gamma (2 + 2) \ Gamma (15 + 40-2)}

,

in which

{\ displaystyle C = {\ frac {\ Gamma (2 + 15)} {\ Gamma (2) \ Gamma (15) \ Gamma (2 + 15 + 40)}}}

and there and besides is general , one obtains ${\ displaystyle {40 \ choose 2} = 780}$ ${\ displaystyle \ Gamma (k) = (k-1)! \,}$

The random variables used in the example

{\ displaystyle {\ begin {aligned} P (X = 2 | n = 40, a = 2, b = 15) & = {\ frac {16!} {1 \ cdot 14! \ cdot 56!}} (780 \ cdot 6 \ cdot 52!) \\ & = 780 \ cdot 6 \ cdot {\ frac {16!} {14!}} \ cdot {\ frac {54!} {56!}} = {\ frac {780 } {53}} \ cdot {\ frac {6} {54}} \ cdot {\ frac {15} {55}} \ cdot {\ frac {16} {56}} \\ & = {\ frac {260 } {53}} \ cdot {\ frac {2} {77}} = 0 {,} 12741975 = 12 {,} 74 \, \%. \ End {aligned}}}

This result differs significantly from that which would have been calculated with a "simple" binomial distribution . In that case the result would be . ${\ displaystyle B (n = 40, p = {\ tfrac {1} {15}})}$ ${\ displaystyle P (X = 2, n = 40, p = {\ tfrac {1} {15}}) = 25 {,} 19 \, \%}$

The graph shows that the “simple” binomial distribution “allows” fewer results than that . This happens because one does not neglect in the Bayesian model that the “true” proportion of red balls is basically unknown, and thus the results are more widely spread. ${\ displaystyle B (n = 40, p = {\ tfrac {1} {15}})}$ ${\ displaystyle BeB (n = 40, a = 2, b = 15)}$

literature

Leonhard Held: Methods of statistical inference. Likelihood and Bayes , with the assistance of Daniel Sabanés Bové, Spektrum Akademischer Verlag Heidelberg 2008, ISBN 978-3-8274-1939-2 .
Jim Albert: Bayesian Computation With R , Springer New York, 2009, ISBN 978-0-387-92297-3 , doi : 10.1007 / 978-0-387-92298-0 .

Web links