dist.Inverse.Wishart {LaplacesDemon}  R Documentation 
Inverse Wishart Distribution
Description
These functions provide the density and random number generation for the inverse Wishart distribution.
Usage
dinvwishart(Sigma, nu, S, log=FALSE)
rinvwishart(nu, S)
Arguments
Sigma 
This is the symmetric, positivedefinite

nu 
This is the scalar degrees of freedom, 
S 
This is the symmetric, positivesemidefinite

log 
Logical. If 
Details
Application: Continuous Multivariate
Density:
p(\theta) = (2^{\nu k/2} \pi^{k(k1)/4} \prod^k_{i=1} \Gamma(\frac{\nu+1i}{2}))^{1} \textbf{S}^{nu/2} \Omega^{(nuk1)/2} \exp(\frac{1}{2} tr(\textbf{S} \Omega^{1}))
Inventor: John Wishart (1928)
Notation 1:
\Sigma \sim \mathcal{W}^{1}_{\nu}(\textbf{S}^{1})
Notation 2:
p(\Sigma) = \mathcal{W}^{1}_{\nu}(\Sigma  \textbf{S}^{1})
Parameter 1: degrees of freedom
\nu
Parameter 2: symmetric, positivesemidefinite
k \times k
scale matrix\textbf{S}
Mean:
E(\Sigma) = \frac{\textbf{S}}{\nu  k  1}
Variance:
Mode:
mode(\Sigma) = \frac{\textbf{S}}{\nu + k + 1}
The inverse Wishart distribution is a probability distribution defined on
realvalued, symmetric, positivedefinite matrices, and is used as the
conjugate prior for the covariance matrix, \Sigma
, of a
multivariate normal distribution. The inverseWishart density is always
finite, and the integral is always finite. A degenerate form occurs when
\nu < k
.
When applicable, the alternative Cholesky parameterization should be
preferred. For more information, see dinvwishartc
.
The inverse Wishart prior lacks flexibility, having only one parameter,
\nu
, to control the variability for all k(k + 1)/2
elements. Popular choices for the scale matrix \textbf{S}
include an identity matrix or sample covariance matrix. When the model
sample size is small, the specification of the scale matrix can be
influential.
The inverse Wishart distribution has a dependency between variance and correlation, although its relative for a precision matrix (inverse covariance matrix), the Wishart distribution, does not have this dependency. This relationship becomes weaker with more degrees of freedom.
Due to these limitations (lack of flexibility, and dependence between
variance and correlation), alternative distributions have been
developed. Alternative distributions that are available here include
HuangWand (dhuangwand
), inverse matrix gamma
(dinvmatrixgamma
), Scaled Inverse Wishart
(dsiw
), and YangBerger (dyangberger
).
These functions are parameterized as per Gelman et al. (2004).
Value
dinvwishart
gives the density and
rinvwishart
generates random deviates.
References
Gelman, A., Carlin, J., Stern, H., and Rubin, D. (2004). "Bayesian Data Analysis, Texts in Statistical Science, 2nd ed.". Chapman and Hall, London.
Wishart, J. (1928). "The Generalised Product Moment Distribution in Samples from a Normal Multivariate Population". Biometrika, 20A(12), p. 32–52.
See Also
dhuangwand
,
dinvmatrixgamma
,
dinvwishartc
,
dmvn
,
dsiw
,
dwishart
, and
dyangberger
.
Examples
library(LaplacesDemon)
x < dinvwishart(matrix(c(2,.3,.3,4),2,2), 3, matrix(c(1,.1,.1,1),2,2))
x < rinvwishart(3, matrix(c(1,.1,.1,1),2,2))