dist.Multivariate.Polya | R Documentation |
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
|
n |
This is the number of random draws to take from the distribution. |
alpha |
This is shape vector |
log |
Logical. If |
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.
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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