R: Multivariate Polya Distribution

dist.Multivariate.Polya {LaplacesDemon} | R Documentation

## 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.

### 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 software@bayesian-inference.com

### 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))

[Package LaplacesDemon version 16.1.1 Index]