rIW: Random Generation from the Conditional Inverse Wishart...

In MCMCglmm: MCMC Generalised Linear Mixed Models

Random Generation from the Conditional Inverse Wishart Distribution

Description

Samples from the inverse Wishart distribution, with the possibility of conditioning on a diagonal submatrix

Usage

rIW(V, nu, fix=NULL, n=1, CM=NULL)

Arguments

V	Expected (co)varaince matrix as nu tends to infinity
nu	degrees of freedom
fix	optional integer indexing the partition to be conditioned on
n	integer: number of samples to be drawn
CM	matrix: optional matrix to condition on. If not given, and fix!=NULL, V_22 is conditioned on

Details

If {\bf W^{-1}} is a draw from the inverse Wishart, fix indexes the diagonal element of {\bf W^{-1}} which partitions {\bf W^{-1}} into 4 submatrices. fix indexes the upper left corner of the lower diagonal matrix and it is this matrix that is conditioned on.

For example partioning {\bf W^{-1}} such that

{\bf W^{-1}} = \left[ \begin{array}{cc} {\bf W^{-1}}_{11}&{\bf W^{-1}}_{12}\\ {\bf W^{-1}}_{21}&{\bf W^{-1}}_{22}\\ \end{array} \right]

fix indexes the upper left corner of {\bf W^{-1}}_{22}. If CM!=NULL then {\bf W^{-1}}_{22} is fixed at CM, otherwise {\bf W^{-1}}_{22} is fixed at \texttt{V}_{22}. For example, if dim(V)=4 and fix=2 then {\bf W^{-1}}_{11} is a 1X1 matrix and {\bf W^{-1}}_{22} is a 3X3 matrix.

Value

if n = 1 a matrix equal in dimension to V, if n>1 a matrix of dimension n x length(V)

Note

In versions of MCMCglmm >1.10 the arguments to rIW have changed so that they are more intuitive in the context of MCMCglmm. Following the notation of Wikipedia (https://en.wikipedia.org/wiki/Inverse-Wishart_distribution) the inverse scale matrix {\bm \Psi}=(\texttt{V*nu}). In earlier versions of MCMCglmm (<1.11) {\bm \Psi} = \texttt{V}^{-1}. Although the old parameterisation is consistent with the riwish function in MCMCpack and the rwishart function in bayesm it is inconsistent with the prior definition for MCMCglmm. The following pieces of code are sampling from the same distributions:

riwish(nu, nu*V)	from MCMCpack
rwishart(nu, solve(nu*V))$IW	from bayesm
rIW(nu, solve(nu*V))	from MCMCglmm <1.11
rIW(V, nu)	from MCMCglmm >=1.11

Author(s)

Jarrod Hadfield [e-mail]

References

Korsgaard, I.R. et. al. 1999 Genetics Selection Evolution 31 (2) 177:181

See Also

rwishart, rwish

Examples

nu<-10
V<-diag(4)
rIW(V, nu, fix=2)

MCMCglmm documentation built on July 9, 2023, 5:24 p.m.