# rIW

##### Random Generation from the Conditional Inverse Wishart Distribution

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

- Keywords
- distribution

##### 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 (http://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 |

##### References

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

##### See Also

##### Examples

```
# NOT RUN {
nu<-10
V<-diag(4)
rIW(V, nu, fix=2)
# }
```

*Documentation reproduced from package MCMCglmm, version 2.29, License: GPL (>= 2)*