Beta-binomial distribution

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In theory and probability statistics, the beta-binomial distribution is the family of discrete probability distributions on the finite support of non-negative integers that arise when the probability of success in each fixed number or known Bernoulli trials is unknown or random. The beta-binomial distribution is a binomial distribution in which the probability of success in each experiment is not fixed but random and follows the beta distribution. It's often used in Bayesian statistics, Bayes empirical methods and classical statistics to capture redundant deployment in binomial distributed data types.

This reduces to Bernoulli's distribution as a special case when n Ã, = Ã, 1. For ? Ã, = Ã, ? Ã, = Ã, 1, it is a discrete uniform distribution from 0 to n . It also approximates arbitrary binomial distribution well for large ? and ? . Beta-binomial is a one-dimensional version of the Dirichlet-multinomial distribution, as the binomial and beta distributions are the univariate versions of the multinomial and Dirichlet distributions, respectively.

Video Beta-binomial distribution

Motivation and derivation

As a merged distribution

Beta Distribution is the conjugate distribution of the binomial distribution. This fact leads to the distribution of a workable analytic compound in which one can assume ${\ displaystyle p}$ parameters in the binomial distribution as taken at random from the beta distribution. That is, if

${\ displaystyle {\ begin {aligned} X & amp; \ sim \ operatorname {Bin} (n, p) \\\ end {aligned}}}$

chemistry

${\ displaystyle {\ begin {aligned} P (X = k | p, n) & amp; = L (p | k) = {n \ pilih k} p ^ {k} (1-p) ^ {nk} \ end {aligned}}} Â Â$

di mana Bin (n, p) singkatan dari distribusi binomial, dan di mana p adalah variabel acak dengan distribusi beta.

${\ displaystyle {\ begin {aligned} \ pi (p | \ alpha, \ beta) & amp; = \ mathrm {Beta} (\ alpha, \ beta) \\ & amp ; = {\ frac {p ^ {\ alpha -1} (1-p) ^ {\ beta -1}} {\ mathrm {B} (\ alpha, \ beta)}} \ end {aligned}}}  ÂÂ$

Menggunakan properti will have a beta fungi, it gives a dry bergantian dish

${\ displaystyle f (k | n, \ alpha, \ beta) = {\ frac {\ Gamma (n 1)} {\ Gamma (k 1) \ Gamma (nk 1)}} {\ frac {\ Gamma (k \ alpha) \ Gamma (nk \ beta)} {\ Gamma (n \ alpha \ beta)}} {\ frac {\ Gamma (\ alpha \ beta )} {\ Gamma (\ alpha) \ Gamma (\ beta)}}.} Â Â$

Beta-binomial sebagai model guci

The beta-binomial distribution can also be motivated through the jar model for positive integer values? and ?, known as Polya Jar models. In particular, imagine a jar containing? red balls and? a black ball, in which a random draw is made. If a red ball is observed, then two red orbs are returned to the jar. Likewise, if a black ball is drawn, then two black balls are returned to the jar. If this is repeated, then the probability of observing the red ball follows the beta-binomial distribution with the parameter n,? and?

Note that if a random draw with a simple replacement (no balls above and above the observed ball is added to the jar), then the distribution follows the binomial distribution and if the random draw is made without replacement, the distribution follows the hypergeometric distribution.

Maps Beta-binomial distribution

Moments and properties

Membiarkan ${\ displaystyle \ pi = {\ frac {\ alpha} {\ alpha \ beta}} \!} Â Â$ kami mencatat, dengan saksama, bahwa mean dapat ditulis sebagai

${\ displaystyle \ mu = {\ frac {n \ alpha} {\ alpha \ beta}} = n \ pi \!} Â Â$

give severalinya sebagai

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di mana ${\ displaystyle \ rho = {\ tfrac {1} {\ alpha \ beta 1}} \!} Â Â$ . Parameter ${\ displaystyle \ rho \!} Â$ dikenal sebagai korelasi "intra class" atau "intra cluster". Ini adalah korelasi positif yang menyebabkan overdispersion.

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Taksiran titik

Metode momen

Metode perkiraan momen dapat diperoleh dengan mencatat momen pertama dan kedua dari beta-binomial yaitu

${\ displaystyle {\ begin {aligned} \ mu_ {1} & amp; = {\ frac {n \ alpha} {\ alpha \ beta}} \\\ mu _ {2} & amp; = {\ frac {n \ alpha [n (1 \ alpha) \ beta]} {(\ alpha \ beta) (1 \ alpha \ beta)}} \ end {selaras }}}  ÂÂ$

dan pengaturan momen mentah ini sama dengan momen sampel mentah pertama dan kedua masing-masing

${\ displaystyle {\ begin {aligned} {\ hat {\ mu}} _ {1} & amp;: = m_ {1} = {\ frac {1} {N }} \ jumlah _ {i = 1} ^ {N} X_ {i} \\ {\ hat {\ mu}} _ {2} & amp;: = m_ {2} = {\ frac {1} {N} } \ jumlah _ {i = 1} ^ {N} X_ {i} ^ {2} \ end {aligned}}}  ÂÂ$

Note that these estimates can be non-sensuously negative which is evidence that the data is not dispersed or less distributed relative to the binomial distribution. In this case, the binometer distribution and hypergeometric distribution are the respective alternative candidates.

Estimated maximum likelihood

While the approximate maximum likelihood of closed forms is impractical, given that pdf consists of common functions (gamma functions and/or Beta functions), they can be easily found through direct numerical optimization. The approximate maximum likelihood of empirical data can be calculated using the general method for the installation of the PÃÆ'³lya multinomial distribution, the method described in (Minka 2003). The VGAM R package through the vglm function, through maximum likelihood, facilitates the installation of glm type models with responses distributed according to the beta-binomial distribution. Note also that there is no requirement that n remains in all observations.

Example

The following data gives the number of boys among the first 12 children of a family of size 13 in 6115 families taken from records of hospitals in the 19th century Saxony (Sokal and Rohlf, p.Ã, 59 from Lindsey). The 13th child is neglected to reduce the influence of families that do not stop randomly when the desired gender is achieved.

Kami mentatat second momen sampel pertama

${\ displaystyle {\ begin {aligned} m1 & amp; = 6.23 \\ m_2 & amp; = 42.31 \\ n & amp; = 12 \ end {aligned}}} Â Â$

dan karena itu metode perkiraan momen adalah

Perkiraan kemungkinan maksimum dapat ditemukan secara numerik

Source of the article : Wikipedia