These functions provide the density and random number generation for the multivariate t distribution, otherwise called the multivariate Student distribution. These functions use the precision and Cholesky parameterization.
This is either a vector of length k or a matrix with a number of columns, k, equal to the number of columns in precision matrix Omega.
This is the number of random draws.
This is a numeric vector representing the location parameter,
mu (the mean vector), of the multivariate distribution
(equal to the expected value when
This is a k x k upper-triangular of the precision matrix that is Cholesky fator U of precision matrix Omega.
This is the degrees of freedom nu, which must be positive.
Application: Continuous Multivariate
p(theta) = (Gamma((nu+k)/2) / (Gamma(nu/2)*nu^(k/2)*pi^(k/2))) * |Omega|^(1/2) * (1 + (1/nu) (theta-mu)^T Omega (theta-mu))^(-(nu+k)/2)
Inventor: Unknown (to me, anyway)
Notation 1: theta ~ t[k](mu, Omega^(-1), nu)
Notation 2: p(theta) = t[k](theta | mu, Omega^(-1), ν)
Parameter 1: location vector mu
Parameter 2: positive-definite k x k precision matrix Omega
Parameter 3: degrees of freedom nu > 0
Mean: E(theta) = mu, for nu > 1, otherwise undefined
Variance: var(theta) = (nu / (nu - 2))*Omega^(-1), for nu> 2
Mode: mode(theta) = mu
The multivariate t distribution, also called the multivariate Student or multivariate Student t distribution, is a multidimensional extension of the one-dimensional or univariate Student t distribution. A random vector is considered to be multivariate t-distributed if every linear combination of its components has a univariate Student t-distribution.
It is usually parameterized with mean and a covariance matrix, or in
Bayesian inference, with mean and a precision matrix, where the
precision matrix is the matrix inverse of the covariance matrix. These
functions provide the precision parameterization for convenience and
familiarity. It is easier to calculate a multivariate t density
with the precision parameterization, because a matrix inversion can be
avoided. The precision matrix is replaced with an upper-triangular
k x k matrix that is Cholesky factor
U, as per the
chol function for Cholesky
This distribution has a mean parameter vector mu of length k, and a k x k precision matrix Omega, which must be positive-definite. When degrees of freedom nu=1, this is the multivariate Cauchy distribution.
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. Overall, the Cholesky
parameterization is faster than the traditional parameterization.
dmvtpc must additionally
matrix-multiply the Cholesky back to the precision matrix, but it
does not have to check for or correct the precision matrix to
positive-definiteness, which overall is slower. Compared with
rmvtpc is faster because the Cholesky decomposition
has already been performed.
dmvtpc gives the density and
rmvtpc generates random deviates.
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