Covariance Matrix Distributions
The covariance matrix distributions have support on symmetric, positive-definite \(K \times K\) matrices or their Cholesky factors (square, lower triangular matrices with positive diagonal elements).
Wishart distribution
Probability density function
If \(K \in \mathbb{N}\), \(\nu \in (K-1,\infty)\), and \(S \in \mathbb{R}^{K \times K}\) is symmetric and positive definite, then for symmetric and positive-definite \(W \in \mathbb{R}^{K \times K}\), \[\begin{equation*} \text{Wishart}(W \mid \nu,S) = \frac{1}{2^{\nu K / 2}} \ \frac{1}{\Gamma_K \! \left( \frac{\nu}{2} \right)} \ \left| S \right|^{-\nu/2} \ \left| W \right|^{(\nu - K - 1)/2} \ \exp \! \left(- \frac{1}{2} \ \text{tr}\left( S^{-1} W \right) \right) \! , \end{equation*}\] where \(\text{tr}()\) is the matrix trace function, and \(\Gamma_K()\) is the multivariate Gamma function, \[\begin{equation*} \Gamma_K(x) = \frac{1}{\pi^{K(K-1)/4}} \ \prod_{k=1}^K \Gamma \left( x + \frac{1 - k}{2} \right) \!. \end{equation*}\]
Distribution statement
W ~ wishart(nu, Sigma)
Increment target log probability density with wishart_lupdf(W | nu, Sigma).
Stan functions
real wishart_lpdf(matrix W | real nu, matrix Sigma)
Return the log of the Wishart density for symmetric and positive-definite matrix W given degrees of freedom nu and symmetric and positive-definite scale matrix Sigma.
real wishart_lupdf(matrix W | real nu, matrix Sigma)
Return the log of the Wishart density for symmetric and positive-definite matrix W given degrees of freedom nu and symmetric and positive-definite scale matrix Sigma dropping constant additive terms.
matrix wishart_rng(real nu, matrix Sigma)
Generate a Wishart variate with degrees of freedom nu and symmetric and positive-definite scale matrix Sigma; may only be used in transformed data and generated quantities blocks.
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 \[\begin{equation*} L_W \sim \textrm{WishartCholesky}(\nu, L_S) \end{equation*}\] if and only if \[\begin{equation*} W \sim \textrm{Wishart}(\nu, S). \end{equation*}\]
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 \[\begin{equation*} \text{WishartCholesky}(L_W \mid \nu,L_S) = \text{Wishart}(L_W L_W^{\top} \mid \nu,L_S L_S^{\top}) \, \left| J_{f^{-1}} \right|, \end{equation*}\] 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 \[\begin{equation*} \log \left| J_{f^{-1}} \right| = K \log(2) + \sum_{k=1}^K (K - k + 1) \log {(L_W)_{k,\, k}}. \end{equation*}\]
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.
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.
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.
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
Inverse Wishart distribution
Probability density function
If \(K \in \mathbb{N}\), \(\nu \in (K-1,\infty)\), and \(S \in \mathbb{R}^{K \times K}\) is symmetric and positive definite, then for symmetric and positive-definite \(W \in \mathbb{R}^{K \times K}\), \[\begin{equation*} \text{InvWishart}(W \mid \nu,S) = \frac{1}{2^{\nu K / 2}} \ \frac{1}{\Gamma_K \! \left( \frac{\nu}{2} \right)} \ \left| S \right|^{\nu/2} \ \left| W \right|^{-(\nu + K + 1)/2} \ \exp \! \left( - \frac{1}{2} \ \text{tr}(SW^{-1}) \right) \! . \end{equation*}\]
Distribution statement
W ~ inv_wishart(nu, Sigma)
Increment target log probability density with inv_wishart_lupdf(W | nu, Sigma).
Stan functions
real inv_wishart_lpdf(matrix W | real nu, matrix Sigma)
Return the log of the inverse Wishart density for symmetric and positive-definite matrix W given degrees of freedom nu and symmetric and positive-definite scale matrix Sigma.
real inv_wishart_lupdf(matrix W | real nu, matrix Sigma)
Return the log of the inverse Wishart density for symmetric and positive-definite matrix W given degrees of freedom nu and symmetric and positive-definite scale matrix Sigma dropping constant additive terms.
matrix inv_wishart_rng(real nu, matrix Sigma)
Generate an inverse Wishart variate with degrees of freedom nu and symmetric and positive-definite scale matrix Sigma; may only be used in transformed data and generated quantities blocks.
Inverse Wishart distribution, Cholesky Parameterization
The Cholesky parameterization of the inverse 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 \[\begin{equation*} L_W \sim \textrm{InvWishartCholesky}(\nu, L_S) \end{equation*}\] if and only if \[\begin{equation*} W \sim \textrm{InvWishart}(\nu, S). \end{equation*}\]
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 inverse Wishart density is \[\begin{equation*} \text{InvWishartCholesky}(L_W \mid \nu,L_S) = \text{InvWishart}(L_WL_W^{\top} \mid \nu, L_S L_S^{\top}) \, \left| J_{f^{-1}} \right|, \end{equation*}\] 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 \[\begin{equation*} \log \left| J_{f^{-1}} \right| = K \log(2) + \sum_{k=1}^K (K - k + 1) \log {(L_W)_{k,\, k}}. \end{equation*}\]
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.
Stan functions
real inv_wishart_cholesky_lpdf(matrix L_W | real nu, matrix L_S)
Return the log of the inverse 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.
real inv_wishart_cholesky_lupdf(matrix L_W | real nu, matrix L_S)
Return the log of the inverse 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.
matrix inv_wishart_cholesky_rng(real nu, matrix L_S)
Generate the Cholesky factor of an inverse 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.