dist.HuangWand: Huang-Wand Distribution
In LaplacesDemon: Complete Environment for Bayesian Inference

Description Usage Arguments Details Value References See Also Examples

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 x k positive-definite covariance matrix Sigma for `dhuangwand`, or the Cholesky factor 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.

Application: Continuous Multivariate
Density: p(theta) = W^(-1)[nu+k-1](2*nu*diag(1/a)) G^(-1)(1/2, 1/A^2)
Inventor: Huang and Wand (2013)
Notation 1: theta ~ HW[nu](a, A)
Notation 2: p(theta) ~ HW[nu](theta | a, 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 HT(nu, 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.

dhuangwand and dhuangwandc give the density, and rhuangwand and rhuangwandc generate random deviates.

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

dhalft and dinvwishart

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