BetaBinom() R function from [extraDistr]

Beta-binomial distribution

Probability mass function and random generation for the beta-binomial distribution.


dbbinom(x, size, alpha = 1, beta = 1, log = FALSE)

pbbinom(q, size, alpha = 1, beta = 1, lower.tail = TRUE, log.p = FALSE)

rbbinom(n, size, alpha = 1, beta = 1)

Arguments

x, q: vector of quantiles.
size: number of trials (zero or more).
alpha, beta: non-negative parameters of the beta distribution.
log, log.p: logical; if TRUE, probabilities p are given as log(p).
lower.tail: logical; if TRUE (default), probabilities are $P[X \le x]$

otherwise, $P[X > x]$ .
n: number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If $p ~ Beta(\alpha, \beta)$ and $X ~ Binomial(n, p)$ , then $X ~ BetaBinomial(n, \alpha, \beta)$ .

Probability mass function

f(x) = {n \choose x} \frac{\mathrm{B}(x+\alpha, n-x+\beta)}{\mathrm{B}(\alpha, \beta)}f(x) = choose(n, x) * B(x+\alpha, n-x+\beta) / B(\alpha, \beta)

Cumulative distribution function is calculated using recursive algorithm that employs the fact that $\Gamma(x) = (x - 1)!$ , and c(" $\n$ ", " $B(x, y) = (\\Gamma(x)\\Gamma(y))/\\Gamma(x+y)\n$ "), and that c(" $\n$ ", " $choose(n,k) = prod((n+1-(1:k))/(1:k))\n$ "). This enables re-writing probability mass function as

f(x) = \left( \prod_{i=1}^x \frac{n+1-i}{i} \right) \frac{\frac{(\alpha+x-1)!\,(\beta+n-x-1)!}{(\alpha+\beta+n-1)!}}{\mathrm{B}(\alpha,\beta)}f(x) = prod((n+1-(1:x))/(1:x)) * (((\alpha+x-1)!*(\beta+n-x-1)!)/((\alpha+\beta+n+1)!)) / B(\alpha, \beta)

what makes recursive updating from $x$ to $x+1$ easy using the properties of factorials

f(x+1) = \left( \prod_{i=1}^x \frac{n+1-i}{i} \right) \frac{n+1-x+1}{x+1} \frac{\frac{(\alpha+x-1)! \,(\alpha+x)\,(\beta+n-x-1)! \, (\beta+n-x)^{-1}}{(\alpha+\beta+n-1)!\,(\alpha+\beta+n)}}{\mathrm{B}(\alpha,\beta)}f(x+1) = prod((n+1-(1:x))/(1:x)) * ((n+1-x+1)/(x+1)) * (((\alpha+x-1)!*(\alpha+x)*(\beta+n-x-1)!/(\beta+n-x))/((\alpha+\beta+n+1)!*(\alpha+\beta+n))) / B(\alpha, \beta)

and let's us efficiently calculate cumulative distribution function as a sum of probability mass functions

F(x) = \sum_{k=0}^x f(k)F(x) = f(0)+...+f(x)

Examples


x <- rbbinom(1e5, 1000, 5, 13)
xx <- 0:1000
hist(x, 100, freq = FALSE)
lines(xx-0.5, dbbinom(xx, 1000, 5, 13), col = "red")
hist(pbbinom(x, 1000, 5, 13))
xx <- seq(0, 1000, by = 0.1)
plot(ecdf(x))
lines(xx, pbbinom(xx, 1000, 5, 13), col = "red", lwd = 2)

BetaBinom function

Beta-binomial distribution

Arguments

Details

Examples

See Also