These are the density and random generation functions for the Huang-Wand prior distribution for a covariance matrix.

```
dhuangwand(x, nu=2, a, A, log=FALSE)
dhuangwandc(x, nu=2, a, A, log=FALSE)
rhuangwand(nu=2, a, A)
rhuangwandc(nu=2, a, A)
```

x

This is a \(k \times k\) positive-definite
covariance matrix \(\Sigma\) for `dhuangwand`

, or the
Cholesky factor \(\textbf{U}\) of the covariance matrix for
`dhuangwandc`

.

nu

This is a scalar degrees of freedom parameter
\(\nu\). The default is `nu=2`

, which is an
uninformative prior, resulting in marginal uniform distributions
on the correlation matrix.

a

This is a positive-only vector of scale parameters \(a\) of length \(k\).

A

This is a positive-only vector of of scale hyperparameters
\(A\) of length \(k\). Larger values result in a more
uninformative prior. A default, uninformative prior is
`A=rep(1e6,k)`

.

log

Logical. If `log=TRUE`

, then the logarithm of the
density is returned.

`dhuangwand`

and `dhuangwandc`

give the density, and
`rhuangwand`

and `rhuangwandc`

generate random deviates.

Application: Continuous Multivariate

Density: \(p(\theta) = \mathcal{W}^{-1}_{\nu+k-1}(2 \nu diag(1/a)) \mathcal{G}^{-1}(1/2, 1/A^2)\)

Inventor: Huang and Wand (2013)

Notation 1: \(\theta \sim \mathcal{HW}_\nu(\textbf{a}, \textbf{A})\)

Notation 2: \(p(\theta) \sim \mathcal{HW}_\nu(\theta | \textbf{a}, \textbf{A})\)

Parameter 1: degrees of freedom \(\nu\)

Parameter 2: scale \(a > 0\)

Parameter 3: scale \(A > 0\)

Mean:

Variance:

Mode:

Huang and Wand (2013) proposed a prior distribution for a covariance matrix that uses a hierarchical inverse Wishart. This is a more flexible alternative to the inverse Wishart distribution, and the Huang-Wand prior retains conjugacy. The Cholesky parameterization is also provided here.

The Huang-Wand prior distribution alleviates two main limitations of an inverse Wishart distribution. First, the uncertainty in the diagonal variances of a covariance matrix that is inverse Wishart distributed is represented with only one degrees of freedom parameter, which may be too restrictive. The Huang-Wand prior overcomes this limitation. Second, the inverse Wishart distribution imposes a dependency between variance and correlation. The Huang-Wand prior lessens, but does not fully remove, this dependency.

The standard deviations of a Huang-Wand distributed covariance matrix are half-t distributed, as \(\mathcal{HT}(\nu, \textbf{A})\). This is in accord with modern assumptions about distributions of scale parameters, and is also useful for sparse covariance matrices.

The `rhuangwand`

function allows either `a`

or `A`

to be
missing. When `a`

is missing, the covariance matrix is generated
from the hyperparameters. When `A`

is missing, the covariance
matrix is generated from the parameters.

Huang, A., Wand, M., et al. (2013), "Simple Marginally Noninformative
Prior Distributions for Covariance Matrices". *Bayesian
Analysis*, 8, p. 439--452.

`dhalft`

and
`dinvwishart`

```
# NOT RUN {
library(LaplacesDemon)
dhuangwand(diag(3), nu=2, a=runif(3), A=rep(1e6,3), log=TRUE)
rhuangwand(nu=2, A=rep(1e6, 3)) #Missing a
rhuangwand(nu=2, a=runif(3)) #Missing A
# }
```

Run the code above in your browser using DataLab