Discrete probability distribution
Probability mass function
Cumulative distribution function
Notation B e t a B i n ( n , α , β ) {\displaystyle \mathrm {BetaBin} (n,\alpha ,\beta )} Parameters n ∈N 0 α > 0 {\displaystyle \alpha >0} real )β > 0 {\displaystyle \beta >0} real )Support x ∈ { 0, …,n }PMF ( n x ) B ( x + α , n − x + β ) B ( α , β ) {\displaystyle {\binom {n}{x}}{\frac {\mathrm {B} (x+\alpha ,n-x+\beta )}{\mathrm {B} (\alpha ,\beta )}}\!} B ( x , y ) = Γ ( x ) Γ ( y ) Γ ( x + y ) {\displaystyle \mathrm {B} (x,y)={\frac {\Gamma (x)\,\Gamma (y)}{\Gamma (x+y)}}} beta function CDF { 0 , x < 0 ( n x ) B ( x + α , n − x + β ) B ( α , β ) 3 F 2 ( a ; b ; x ) , 0 ≤ x < n 1 , x ≥ n {\displaystyle {\begin{cases}0,&x<0\\{\binom {n}{x}}{\tfrac {\mathrm {B} (x+\alpha ,n-x+\beta )}{\mathrm {B} (\alpha ,\beta )}}{}_{3}\!F_{2}({\boldsymbol {a}};{\boldsymbol {b}};x),&0\leq x<n\\1,&x\geq n\end{cases}}} 3 F 2 (a ;b ;x)generalized hypergeometric function 3 F 2 ( 1 , − x , n − x + β ; n − x + 1 , 1 − x − α ; 1 ) {\displaystyle {}_{3}\!F_{2}(1,-x,n\!-\!x\!+\!\beta ;n\!-\!x\!+\!1,1\!-\!x\!-\!\alpha ;1)\!} Mean n α α + β {\displaystyle {\frac {n\alpha }{\alpha +\beta }}\!} Variance n α β ( α + β + n ) ( α + β ) 2 ( α + β + 1 ) {\displaystyle {\frac {n\alpha \beta (\alpha +\beta +n)}{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!} Skewness ( α + β + 2 n ) ( β − α ) ( α + β + 2 ) 1 + α + β n α β ( n + α + β ) {\displaystyle {\tfrac {(\alpha +\beta +2n)(\beta -\alpha )}{(\alpha +\beta +2)}}{\sqrt {\tfrac {1+\alpha +\beta }{n\alpha \beta (n+\alpha +\beta )}}}\!} Excess kurtosis See text MGF 2 F 1 ( − n , α ; α + β ; 1 − e t ) {\displaystyle _{2}F_{1}(-n,\alpha ;\alpha +\beta ;1-e^{t})\!} 2 F 1 {\displaystyle _{2}F_{1}} hypergeometric function CF 2 F 1 ( − n , α ; α + β ; 1 − e i t ) {\displaystyle _{2}F_{1}(-n,\alpha ;\alpha +\beta ;1-e^{it})\!} PGF 2 F 1 ( − n , α ; α + β ; 1 − z ) {\displaystyle _{2}F_{1}(-n,\alpha ;\alpha +\beta ;1-z)\!}
Inprobability theory andstatistics , thebeta-binomial distribution is a family of discreteprobability distributions on a finitesupport of non-negative integers arising when the probability of success in each of a fixed or known number ofBernoulli trials is either unknown or random. The beta-binomial distribution is thebinomial distribution in which the probability of success at each ofn trials is not fixed but randomly drawn from abeta distribution . It is frequently used inBayesian statistics ,empirical Bayes methods andclassical statistics to captureoverdispersion inbinomial type distributed data.
The beta-binomial is a one-dimensional version of theDirichlet-multinomial distribution as the binomial and beta distributions are univariate versions of themultinomial andDirichlet distributions respectively. The special case whereα andβ are integers is also known as thenegative hypergeometric distribution .
Motivation and derivation [ edit ] As a compound distribution [ edit ] Thebeta distribution is aconjugate distribution of thebinomial distribution . This fact leads to an analytically tractablecompound distribution where one can think of thep {\displaystyle p} x {\displaystyle x} n {\displaystyle n}
f ( x ∣ n , α , β ) = ∫ 0 1 B i n ( x | n , p ) B e t a ( p ∣ α , β ) d p = ( n x ) 1 B ( α , β ) ∫ 0 1 p x + α − 1 ( 1 − p ) n − x + β − 1 d p = ( n x ) B ( x + α , n − x + β ) B ( α , β ) . {\displaystyle {\begin{aligned}f(x\mid n,\alpha ,\beta )&=\int _{0}^{1}\mathrm {Bin} (x|n,p)\mathrm {Beta} (p\mid \alpha ,\beta )\,dp\\[6pt]&={n \choose x}{\frac {1}{\mathrm {B} (\alpha ,\beta )}}\int _{0}^{1}p^{x+\alpha -1}(1-p)^{n-x+\beta -1}\,dp\\[6pt]&={n \choose x}{\frac {\mathrm {B} (x+\alpha ,n-x+\beta )}{\mathrm {B} (\alpha ,\beta )}}.\end{aligned}}} Using the properties of thebeta function , this can alternatively be written
f ( x ∣ n , α , β ) = Γ ( n + 1 ) Γ ( x + α ) Γ ( n − x + β ) Γ ( n + α + β ) Γ ( x + 1 ) Γ ( n − x + 1 ) Γ ( α + β ) Γ ( α ) Γ ( β ) {\displaystyle f(x\mid n,\alpha ,\beta )={\frac {\Gamma (n+1)\Gamma (x+\alpha )\Gamma (n-x+\beta )}{\Gamma (n+\alpha +\beta )\Gamma (x+1)\Gamma (n-x+1)}}{\frac {\Gamma (\alpha +\beta )}{\Gamma (\alpha )\Gamma (\beta )}}} The beta-binomial distribution can also be motivated via anurn model for positiveinteger values ofα andβ , known as thePólya urn model . Specifically, imagine an urn containingα red balls andβ black balls, where random draws are made. If a red ball is observed, then two red balls are returned to the urn. Likewise, if a black ball is drawn, then two black balls are returned to the urn. If this is repeatedn times, then the probability of observingx red balls follows a beta-binomial distribution with parametersn ,α and β .
By contrast, if the random draws are with simple replacement (no balls over and above the observed ball are added to the urn), then the distribution follows a binomial distribution and if the random draws are made without replacement, the distribution follows ahypergeometric distribution .
Moments and properties [ edit ] The first three rawmoments are
μ 1 = n α α + β μ 2 = n α [ n ( 1 + α ) + β ] ( α + β ) ( 1 + α + β ) μ 3 = n α [ n 2 ( 1 + α ) ( 2 + α ) + 3 n ( 1 + α ) β + β ( β − α ) ] ( α + β ) ( 1 + α + β ) ( 2 + α + β ) {\displaystyle {\begin{aligned}\mu _{1}&={\frac {n\alpha }{\alpha +\beta }}\\[8pt]\mu _{2}&={\frac {n\alpha [n(1+\alpha )+\beta ]}{(\alpha +\beta )(1+\alpha +\beta )}}\\[8pt]\mu _{3}&={\frac {n\alpha [n^{2}(1+\alpha )(2+\alpha )+3n(1+\alpha )\beta +\beta (\beta -\alpha )]}{(\alpha +\beta )(1+\alpha +\beta )(2+\alpha +\beta )}}\end{aligned}}} and thekurtosis is
β 2 = ( α + β ) 2 ( 1 + α + β ) n α β ( α + β + 2 ) ( α + β + 3 ) ( α + β + n ) [ ( α + β ) ( α + β − 1 + 6 n ) + 3 α β ( n − 2 ) + 6 n 2 − 3 α β n ( 6 − n ) α + β − 18 α β n 2 ( α + β ) 2 ] . {\displaystyle \beta _{2}={\frac {(\alpha +\beta )^{2}(1+\alpha +\beta )}{n\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)(\alpha +\beta +n)}}\left[(\alpha +\beta )(\alpha +\beta -1+6n)+3\alpha \beta (n-2)+6n^{2}-{\frac {3\alpha \beta n(6-n)}{\alpha +\beta }}-{\frac {18\alpha \beta n^{2}}{(\alpha +\beta )^{2}}}\right].} Lettingp = α α + β {\displaystyle p={\frac {\alpha }{\alpha +\beta }}\!}
μ = n α α + β = n p {\displaystyle \mu ={\frac {n\alpha }{\alpha +\beta }}=np\!} and the variance as
σ 2 = n α β ( α + β + n ) ( α + β ) 2 ( α + β + 1 ) = n p ( 1 − p ) α + β + n α + β + 1 = n p ( 1 − p ) [ 1 + ( n − 1 ) ρ ] {\displaystyle \sigma ^{2}={\frac {n\alpha \beta (\alpha +\beta +n)}{(\alpha +\beta )^{2}(\alpha +\beta +1)}}=np(1-p){\frac {\alpha +\beta +n}{\alpha +\beta +1}}=np(1-p)[1+(n-1)\rho ]\!} whereρ = 1 α + β + 1 {\displaystyle \rho ={\tfrac {1}{\alpha +\beta +1}}\!} ρ {\displaystyle \rho \;\!} n = 1 {\displaystyle n=1}
Ther factorial moment of a Beta-binomial random variableX
E [ ( X ) r ] = n ! ( n − r ) ! B ( α + r , β ) B ( α , β ) = ( n ) r B ( α + r , β ) B ( α , β ) {\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}={\frac {n!}{(n-r)!}}{\frac {B(\alpha +r,\beta )}{B(\alpha ,\beta )}}=(n)_{r}{\frac {B(\alpha +r,\beta )}{B(\alpha ,\beta )}}} Themethod of moments estimates can be gained by noting the first and second moments of the beta-binomial and setting those equal to the sample momentsm 1 {\displaystyle m_{1}} m 2 {\displaystyle m_{2}}
α ^ = n m 1 − m 2 n ( m 2 m 1 − m 1 − 1 ) + m 1 β ^ = ( n − m 1 ) ( n − m 2 m 1 ) n ( m 2 m 1 − m 1 − 1 ) + m 1 . {\displaystyle {\begin{aligned}{\widehat {\alpha }}&={\frac {nm_{1}-m_{2}}{n({\frac {m_{2}}{m_{1}}}-m_{1}-1)+m_{1}}}\\[5pt]{\widehat {\beta }}&={\frac {(n-m_{1})(n-{\frac {m_{2}}{m_{1}}})}{n({\frac {m_{2}}{m_{1}}}-m_{1}-1)+m_{1}}}.\end{aligned}}} These estimates can be non-sensically negative which is evidence that the data is either undispersed or underdispersed relative to the binomial distribution. In this case, the binomial distribution and thehypergeometric distribution are alternative candidates respectively.
Maximum likelihood estimation [ edit ] While closed-formmaximum likelihood estimates are impractical, given that the pdf consists of common functions (gamma function and/or Beta functions), they can be easily found via direct numerical optimization. Maximum likelihood estimates from empirical data can be computed using general methods for fitting multinomial Pólya distributions, methods for which are described in(Minka 2003). TheR package VGAM through the function vglm, via maximum likelihood, facilitates the fitting ofglm type models with responses distributed according to the beta-binomial distribution. There is no requirement that n is fixed throughout the observations.
Example: Sex ratio heterogeneity [ edit ] The following data gives the number of male children among the first 12 children of family size 13 in 6115 families taken from hospital records in 19th centurySaxony (Sokal and Rohlf, p. 59 from Lindsey). The 13th child is ignored to blunt the effect of families non-randomly stopping when a desired gender is reached.
Males 0 1 2 3 4 5 6 7 8 9 10 11 12 Families 3 24 104 286 670 1033 1343 1112 829 478 181 45 7
The first two sample moments are
m 1 = 6.23 m 2 = 42.31 n = 12 {\displaystyle {\begin{aligned}m_{1}&=6.23\\m_{2}&=42.31\\n&=12\end{aligned}}} and therefore the method of moments estimates are
α ^ = 34.1350 β ^ = 31.6085. {\displaystyle {\begin{aligned}{\widehat {\alpha }}&=34.1350\\{\widehat {\beta }}&=31.6085.\end{aligned}}} Themaximum likelihood estimates can be found numerically
α ^ m l e = 34.09558 β ^ m l e = 31.5715 {\displaystyle {\begin{aligned}{\widehat {\alpha }}_{\mathrm {mle} }&=34.09558\\{\widehat {\beta }}_{\mathrm {mle} }&=31.5715\end{aligned}}} and the maximized log-likelihood is
log L = − 12492.9 {\displaystyle \log {\mathcal {L}}=-12492.9} from which we find theAIC
A I C = 24989.74. {\displaystyle {\mathit {AIC}}=24989.74.} The AIC for the competing binomial model is AIC = 25070.34 and thus we see that the beta-binomial model provides a superior fit to the data i.e. there is evidence for overdispersion.Trivers and Willard postulate a theoretical justification for heterogeneity in gender-proneness amongmammalian offspring.
The superior fit is evident especially among the tails
Males 0 1 2 3 4 5 6 7 8 9 10 11 12 Observed Families 3 24 104 286 670 1033 1343 1112 829 478 181 45 7 Fitted Expected (Beta-Binomial) 2.3 22.6 104.8 310.9 655.7 1036.2 1257.9 1182.1 853.6 461.9 177.9 43.8 5.2 Fitted Expected (Binomialp = 0.519215) 0.9 12.1 71.8 258.5 628.1 1085.2 1367.3 1265.6 854.2 410.0 132.8 26.1 2.3
Role in Bayesian statistics [ edit ] The beta-binomial distribution plays a prominent role in the Bayesian estimation of a Bernoulli success probabilityp {\displaystyle p} X = { X 1 , X 2 , ⋯ X n 1 } {\displaystyle \mathbf {X} =\{X_{1},X_{2},\cdots X_{n_{1}}\}} sample ofindependent and identically distributed Bernoulli random variablesX i ∼ Bernoulli ( p ) {\displaystyle X_{i}\sim {\text{Bernoulli}}(p)} p {\displaystyle p} prior distribution p ∼ Beta ( α , β ) {\displaystyle p\sim {\text{Beta}}(\alpha ,\beta )} Y 1 = ∑ i = 1 n 1 X i {\displaystyle Y_{1}=\sum _{i=1}^{n_{1}}X_{i}} compounding , the prior predictive distribution of
Y 1 ∼ BetaBin ( n 1 , α , β ) {\displaystyle Y_{1}\sim {\text{BetaBin}}(n_{1},\alpha ,\beta )} After observingY 1 {\displaystyle Y_{1}} posterior distribution forp {\displaystyle p}
f ( p | X , α , β ) ∝ ( ∏ i = 1 n 1 p x i ( 1 − p ) 1 − x i ) p α − 1 ( 1 − p ) β − 1 = C p ∑ x i + α − 1 ( 1 − p ) n 1 − ∑ x i + β − 1 = C p y 1 + α − 1 ( 1 − p ) n 1 − y 1 + β − 1 {\displaystyle {\begin{aligned}f(p|\mathbf {X} ,\alpha ,\beta )&\propto \left(\prod _{i=1}^{n_{1}}p^{x_{i}}(1-p)^{1-x_{i}}\right)p^{\alpha -1}(1-p)^{\beta -1}\\&=Cp^{\sum x_{i}+\alpha -1}(1-p)^{n_{1}-\sum x_{i}+\beta -1}\\&=Cp^{y_{1}+\alpha -1}(1-p)^{n_{1}-y_{1}+\beta -1}\end{aligned}}} whereC {\displaystyle C} normalizing constant . We recognize the posterior distribution ofp {\displaystyle p} B e t a ( y 1 + α , n 1 − y 1 + β ) {\displaystyle \mathrm {Beta} (y_{1}+\alpha ,n_{1}-y_{1}+\beta )}
Thus, again through compounding, we find that theposterior predictive distribution of a sumY 2 {\displaystyle Y_{2}} n 2 {\displaystyle n_{2}} B e r n o u l l i ( p ) {\displaystyle \mathrm {Bernoulli} (p)}
Y 2 ∼ B e t a B i n ( n 2 , y 1 + α , n 1 − y 1 + β ) {\displaystyle Y_{2}\sim \mathrm {BetaBin} (n_{2},y_{1}+\alpha ,n_{1}-y_{1}+\beta )} Generating random variates [ edit ] To draw a beta-binomial random variateX ∼ B e t a B i n ( n , α , β ) {\displaystyle X\sim \mathrm {BetaBin} (n,\alpha ,\beta )} p ∼ B e t a ( α , β ) {\displaystyle p\sim \mathrm {Beta} (\alpha ,\beta )} X ∼ B ( n , p ) {\displaystyle X\sim \mathrm {B} (n,p)}
Related distributions [ edit ] B e t a B i n ( 1 , α , β ) ∼ B e r n o u l l i ( p ) {\displaystyle \mathrm {BetaBin} (1,\alpha ,\beta )\sim \mathrm {Bernoulli} (p)\,} p = α α + β {\displaystyle p={\frac {\alpha }{\alpha +\beta }}\,} B e t a B i n ( n , 1 , 1 ) ∼ U ( 0 , n ) {\displaystyle \mathrm {BetaBin} (n,1,1)\sim U(0,n)\,} U ( a , b ) {\displaystyle U(a,b)\,} discrete uniform distribution .IfX ∼ B e t a B i n ( n , α , β ) {\displaystyle X\sim \mathrm {BetaBin} (n,\alpha ,\beta )\,} ( n − X ) ∼ B e t a B i n ( n , β , α ) {\displaystyle (n-X)\sim \mathrm {BetaBin} (n,\beta ,\alpha )\,} lim s → ∞ B e t a B i n ( n , p s , ( 1 − p ) s ) ∼ B ( n , p ) {\displaystyle \lim _{s\rightarrow \infty }\mathrm {BetaBin} (n,ps,(1-p)s)\sim \mathrm {B} (n,p)\,} p = α α + β {\displaystyle p={\frac {\alpha }{\alpha +\beta }}\,} s = α + β {\displaystyle s=\alpha +\beta \,} B ( n , p ) {\displaystyle \mathrm {B} (n,p)\,} binomial distribution .lim n → ∞ B e t a B i n ( n , n λ , n 2 ) ∼ P o i s ( λ ) {\displaystyle \lim _{n\rightarrow \infty }\mathrm {BetaBin} (n,n\lambda ,n^{2})\sim \mathrm {Pois} (\lambda )\,} P o i s ( λ ) {\displaystyle \mathrm {Pois} (\lambda )\,} Poisson distribution .lim n → ∞ B e t a B i n ( n , 1 , n p ( 1 − p ) ) ∼ G e o m ( p ) {\displaystyle \lim _{n\rightarrow \infty }\mathrm {BetaBin} (n,1,{\frac {np}{(1-p)}})\sim \mathrm {Geom} (p)\,} G e o m ( p ) {\displaystyle \mathrm {Geom} (p)\,} geometric distribution .lim n → ∞ B e t a B i n ( n , r , n p ( 1 − p ) ) ∼ N B ( r , p ) {\displaystyle \lim _{n\rightarrow \infty }\mathrm {BetaBin} (n,r,{\frac {np}{(1-p)}})\sim \mathrm {NB} (r,p)\,} N B ( r , p ) {\displaystyle \mathrm {NB} (r,p)\,} negative binomial distribution .
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
