Multivariate Polya Distribution
These functions provide the density and random number generation for the multivariate Polya distribution.
dmvpolya(x, alpha, log=FALSE) rmvpolya(n, alpha)
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 |
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.
dmvpolya gives the density and rmvpolya generates random
deviates.
Statisticat, LLC software@bayesian-inference.com
dcat,
ddirichlet, and
dmultinom.
library(LaplacesDemon) dmvpolya(x=1:3, alpha=1:3, log=TRUE) x <- rmvpolya(1000, c(0.1,0.3,0.6))
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