LaplacesDemon (version 16.1.0)

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))
# }