These functions provide the density and random number generation for the inverse Wishart distribution.

```
dinvwishart(Sigma, nu, S, log=FALSE)
rinvwishart(nu, S)
```

Sigma

This is the symmetric, positive-definite \(k \times k\) matrix \(\Sigma\).

nu

This is the scalar degrees of freedom, \(\nu\).

S

This is the symmetric, positive-semidefinite \(k \times k\) scale matrix \(\textbf{S}\).

log

Logical. If `log=TRUE`

, then the logarithm of the
density is returned.

`dinvwishart`

gives the density and
`rinvwishart`

generates random deviates.

Application: Continuous Multivariate

Density: \(p(\theta) = (2^{\nu k/2} \pi^{k(k-1)/4} \prod^k_{i=1} \Gamma(\frac{\nu+1-i}{2}))^{-1} |\textbf{S}|^{nu/2} |\Omega|^{-(nu-k-1)/2} \exp(-\frac{1}{2} tr(\textbf{S} \Omega^{-1}))\)

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, positive-semidefinite \(k \times k\) 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 real-valued, symmetric, positive-definite matrices, and is used as the conjugate prior for the covariance matrix, \(\Sigma\), of a multivariate normal distribution. The inverse-Wishart 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
Huang-Wand (`dhuangwand`

), inverse matrix gamma
(`dinvmatrixgamma`

), Scaled Inverse Wishart
(`dsiw`

), and Yang-Berger (`dyangberger`

).

These functions are parameterized as per Gelman et al. (2004).

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(1-2), p. 32--52.

`dhuangwand`

,
`dinvmatrixgamma`

,
`dinvwishartc`

,
`dmvn`

,
`dsiw`

,
`dwishart`

, and
`dyangberger`

.

```
# NOT RUN {
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))
# }
```

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