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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