28.2 Wishart distribution, Cholesky Parameterization
The Cholesky parameterization of the Wishart distribution uses a Cholesky factor for both the variate and the parameter. If \(S\) and \(W\) are positive definite matrices with Cholesky factors \(L_S\) and \(L_W\) (i.e., \(S = L_S L_S^{\top}\) and \(W = L_W L_W^{\top}\)), then the Cholesky parameterization is defined so that \[ L_W \sim \textrm{WishartCholesky}(\nu, L_S) \] if and only if \[ W \sim \textrm{Wishart}(\nu, S). \]
28.2.1 Probability density function
If \(K \in \mathbb{N}\), \(\nu \in (K-1, \infty)\), and \(L_S, L_W \in \mathbb{R}^{K \times K}\) are lower triangular matrixes with positive diagonal elements, then the Cholesky parameterized Wishart density is \[ \text{WishartCholesky}(L_W \mid \nu,L_S) = \text{Wishart}(L_W L_W^{\top} \mid \nu,L_S L_S^{\top}) \, \left| J_{f^{-1}} \right|, \] where \(J_{f^{-1}}\) is the Jacobian of the (inverse) transform of the variate, \(f^{-1}(L_W) = L_W L_W^{\top}\). The log absolute determinant is \[ \log \left| J_{f^{-1}} \right| = K \log(2) \sum_{k=1}^K (K - k + 1) \log {L_{W_{k,\, k}}}. \]
The probability functions will raise errors if \(\nu \leq K - 1\) or if \(L_S\) and \(L_W\) are not Cholesky factors (square, lower-triangular matrices with positive diagonal elements) of the same size.
28.2.2 Stan functions
real wishart_cholesky_lpdf(matrix L_W | real nu, matrix L_S)
Return the log of the Wishart density for lower-triangular Cholesky factor L_W
given degrees of freedom nu and lower-triangular Cholesky factor of the
scale matrix L_S.
Available since 2.30
real wishart_cholesky_lupdf(matrix L_W | real nu, matrix L_S)
Return the log of the Wishart density for lower-triangular Cholesky factor of L_W
given degrees of freedom nu and lower-triangular Cholesky factor of the
scale matrix L_S dropping constant additive terms.
Available since 2.30
matrix wishart_cholesky_rng(real nu, matrix L_S)
Generate the Cholesky factor of a Wishart variate with degrees of freedom nu and
lower-triangular Cholesky factor of the scale matrix L_S; may only be used in
transformed data and generated quantities blocks
Available since 2.30