LaplacesDemon (version 16.1.1)

dist.Multivariate.Polya: Multivariate Polya Distribution

Description

These functions provide the density and random number generation for the multivariate Polya distribution.

Usage

dmvpolya(x, alpha, log=FALSE)
rmvpolya(n, alpha)

Arguments

x

This is data or parameters in the form of a vector of length \(k\).

n

This is the number of random draws to take from the distribution.

alpha

This is shape vector \(\alpha\) with length \(k\).

log

Logical. If log=TRUE, then the logarithm of the density is returned.

Value

dmvpolya gives the density and rmvpolya generates random deviates.

Details

• Application: Discrete Multivariate

• Density: $$p(\theta) = \frac{N!}{\prod_k N_k!} \frac{(\sum_k \alpha_k - 1)!}{(\sum_k \theta_k + \sum_k \alpha_k - 1)!} \frac{\prod (\theta + \alpha - 1)!}{(\alpha - 1)!}$$

• Inventor: George Polya (1887-1985)

• Notation 1: \(\theta \sim \mathcal{MPO}(\alpha)\)

• Notation 3: \(p(\theta) = \mathcal{MPO}(\theta | \alpha)\)

• Parameter 1: shape parameter vector \(\alpha\)

• Mean: \(E(\theta) = \)

• Variance: \(var(\theta) =\)

• Mode: \(mode(\theta) = \)

The multivariate Polya distribution is named after George Polya (1887-1985). It is also called the Dirichlet compound multinomial distribution or the Dirichlet-multinomial distribution. The multivariate Polya distribution is a compound probability distribution, where a probability vector \(p\) is drawn from a Dirichlet distribution with parameter vector \(\alpha\), and a set of \(N\) discrete samples is drawn from the categorical distribution with probability vector \(p\) and having \(K\) discrete categories. The compounding corresponds to a Polya urn scheme. In document classification, for example, the distribution is used to represent probabilities over word counts for different document types. The multivariate Polya distribution is a multivariate extension of the univariate Beta-binomial distribution.

See Also

dcat, ddirichlet, and dmultinom.

Examples

# NOT RUN {
library(LaplacesDemon)
dmvpolya(x=1:3, alpha=1:3, log=TRUE)
x <- rmvpolya(1000, c(0.1,0.3,0.6))
# }