dist.HuangWand {LaplacesDemon}

Huang-Wand Distribution

Description

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

Usage

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)

Arguments

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.

Details

• 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.

Value

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

References

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

See Also

dhalft and dinvwishart

Examples

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