Compute the density of an observation of a random Wishart distributed matrix
(dWishart) or an observation
from the inverse Wishart distribution (dInvWishart).
dWishart(x, df, Sigma, log = TRUE)
dInvWishart(x, df, Sigma, log = TRUE)positive definite \(p \times p\) observations for density estimation - either one matrix or a 3-D array.
numeric parameter, "degrees of freedom".
positive definite \(p \times p\) "scale" matrix, the matrix parameter of the distribution.
logical, whether to return value on the log scale.
Density or log of density
Note there are different ways of parameterizing the Inverse Wishart distribution, so check which one you need. Here, If \(X \sim IW_p(\Sigma, \nu)\) then \(X^{-1} \sim W_p(\Sigma^{-1}, \nu)\). Dawid (1981) has a different definition: if \(X \sim W_p(\Sigma^{-1}, \nu)\) and \(\nu > p - 1\), then \(X^{-1} = Y \sim IW(\Sigma, \delta)\), where \(\delta = \nu - p + 1\).
dInvWishart(): density for the inverse Wishart distribution.
Dawid, A. (1981). Some Matrix-Variate Distribution Theory: Notational Considerations and a Bayesian Application. Biometrika, 68(1), 265-274. doi:10.2307/2335827
Gupta, A. K. and D. K. Nagar (1999). Matrix variate distributions. Chapman and Hall.
Mardia, K. V., J. T. Kent, and J. M. Bibby (1979) Multivariate Analysis, London: Academic Press.
set.seed(20180222)
A <- rWishart(1, 10, diag(4))[, , 1]
A
#> [,1] [,2] [,3] [,4]
#> [1,] 8.70459209 -0.21410794 -0.05424324 8.81416710
#> [2,] -0.21410794 10.66557858 5.22611563 -0.09148444
#> [3,] -0.05424324 5.22611563 12.65126374 1.50127740
#> [4,] 8.81416710 -0.09148444 1.50127740 16.84890844
dWishart(x = A, df = 10, Sigma = diag(4L), log = TRUE)
#> [1] -28.14864
dInvWishart(x = solve(A), df = 10, Sigma = diag(4L), log = TRUE)
#> [1] 16.26773