dist.Multivariate.Polya: Multivariate Polya Distribution

In LaplacesDemon: Complete Environment for Bayesian Inference

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.

Details

• Application: Discrete Multivariate

• Density:

  p(theta) = (N! / prod(N[k]!)) * ((sum alpha[k] - 1)! / (sum theta[k] + sum alpha[k] - 1)!) * prod((theta + alpha - 1)! / (alpha - 1)!)

• Inventor: George Polya (1887-1985)

• Notation 1: theta ~ MPO(alpha)

• Notation 3: p(theta) = 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.

Value

dmvpolya gives the density and rmvpolya generates random deviates.

Author(s)

Statisticat, LLC [e-mail]

See Also

dcat, ddirichlet, and dmultinom.

Examples

library(LaplacesDemon)
dmvpolya(x=1:3, alpha=1:3, log=TRUE)
x <- rmvpolya(1000, c(0.1,0.3,0.6))

Example output

[1] -2.734368

LaplacesDemon documentation built on July 9, 2021, 5:07 p.m.