dist.Inverse.Wishart.Cholesky: Inverse Wishart Distribution: Cholesky Parameterization
In benmarwick/LaplacesDemon: Complete Environment for Bayesian Inference

Description Usage Arguments Details Value References See Also Examples

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

1 2	dinvwishartc(U, nu, S, log=FALSE) rinvwishartc(nu, S)

`U`	This is the upper-triangular k x k matrix for the Cholesky factor U of covariance matrix Sigma.
`nu`	This is the scalar degrees of freedom, nu.
`S`	This is the symmetric, positive-semidefinite k x k scale matrix S.
`log`	Logical. If `log=TRUE`, then the logarithm of the density is returned.

Application: Continuous Multivariate
Density: p(theta) = (2^(nu*k/2) * pi^(k(k-1)/4) * [Gamma((nu+1-i)/2) * ... * Gamma((nu+1-k)/2)])^(-1) * |S|^(nu/2) * |Omega|^(-(nu-k-1)/2) * exp(-(1/2) * tr(S Omega^(-1)))
Inventor: John Wishart (1928)
Notation 1: Sigma ~ W^(-1)[nu](S^(-1))
Notation 2: p(Sigma) = W^-1[nu](Sigma | S^(-1))
Parameter 1: degrees of freedom nu
Parameter 2: symmetric, positive-semidefinite k x k scale matrix S
Mean: E(Sigma) = S / (nu - k - 1)
Variance:
Mode: mode(Sigma) = 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. In this parameterization, Sigma has been decomposed to the upper-triangular Cholesky factor U, as per chol. The inverse-Wishart density is always finite, and the integral is always finite. A degenerate form occurs when nu < k.

In practice, U is fully unconstrained for proposals when its diagonal is log-transformed. The diagonal is exponentiated after a proposal and before other calculations. The Cholesky parameterization is faster than the traditional parameterization.

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 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.

One of many alternatives is to use hierarchical priors, in which the main diagonal of the (identity) scale matrix and the degrees of freedom are treated as unknowns (Bouriga and Feron, 2011; Daniels and Kass, 1999). A hierarchical inverse Wishart prior provides shrinkage toward diagonality. Another alternative is to abandon the inverse Wishart distribution altogether for the more flexible method of Barnard et al. (2000) or the horseshoe distribution (dhs) for sparse covariance matrices.

dinvwishartc gives the density and rinvwishartc generates random deviates.

Barnard, J., McCulloch, R., and Meng, X. (2000). "Modeling Covariance Matrices in Terms of Standard Deviations and Correlations, with Application to Shrinkage". Statistica Sinica, 10, p. 1281–1311.

Bouriga, M. and Feron, O. (2011). "Estimation of Covariance Matrices Based on Hierarchical Inverse-Wishart Priors". URL: http://www.citebase.org/abstract?id=oai:arXiv.org:1106.3203.

Daniels, M., and Kass, R. (1999). "Nonconjugate Bayesian Estimation of Covariance Matrices and its use in Hierarchical Models". Journal of the American Statistical Association, 94(448), p. 1254–1263.

Wishart, J. (1928). "The Generalised Product Moment Distribution in Samples from a Normal Multivariate Population". Biometrika, 20A(1-2), p. 32–52.

chol, Cov2Prec, dhs, dmvn, dmvnc, dmvtc, dwishart, and dwishartc.

library(LaplacesDemon)
Sigma <- matrix(c(2,-.3,-.3,4),2,2)
U <- chol(Sigma)
x <- dinvwishartc(U, 3, matrix(c(1,.1,.1,1),2,2))
x <- rinvwishartc(3, matrix(c(1,.1,.1,1),2,2))