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