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 $K\in \mathbb{N}$, $\nu \in (K-1,\mathrm{\infty })$, and $S\in {\mathbb{R}}^{K\times K}$ is symmetric and positive definite, then for symmetric and positive-definite $W\in {\mathbb{R}}^{K\times K}$, $$\text{Wishart}(W\mid \nu ,S)=\frac{1}{2^{\nu K/2}}\frac{1}{{\mathrm{\Gamma }}_K\left(\frac{\nu }{2}\right)}{\left|S\right|}^{-\nu /2}{\left|W\right|}^{(\nu -K-1)/2}\exp (-\frac{1}{2}\text{tr}\left(S^{-1}W\right)),$$ where $\text{tr}()$ is the matrix trace function, and ${\mathrm{\Gamma }}_K()$ is the multivariate Gamma function, $${\mathrm{\Gamma }}_K(x)=\frac{1}{{\pi }^{K(K-1)/4}}\prod _{k=1}^K\mathrm{\Gamma }(x+\frac{1-k}{2}).$$

Distribution statement

W ~ wishart(nu, Sigma)

Increment target log probability density with wishart_lupdf(W | nu, Sigma).

Available since 2.0

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.

Available since 2.12

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.

Available since 2.25

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.

Available since 2.0

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., $S=L_S{L}_S^{\mathrm{\top }}$ and $W=L_W{L}_W^{\mathrm{\top }}$), then the Cholesky parameterization is defined so that $$L_W\sim \text{WishartCholesky}(\nu ,L_S)$$ if and only if $$W\sim \text{Wishart}(\nu ,S).$$

Probability density function

If $K\in \mathbb{N}$, $\nu \in (K-1,\mathrm{\infty })$, 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 $$\text{WishartCholesky}(L_W\mid \nu ,L_S)=\text{Wishart}(L_W{L}_W^{\mathrm{\top }}\mid \nu ,L_S{L}_S^{\mathrm{\top }})\left|J_{f^{-1}}\right|,$$ where $J_{f^{-1}}$ is the Jacobian of the (inverse) transform of the variate, $f^{-1}(L_W)=L_W{L}_W^{\mathrm{\top }}$. 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 $\nu \le K-1$ 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.

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

Inverse Wishart distribution

Probability density function

If $K\in \mathbb{N}$, $\nu \in (K-1,\mathrm{\infty })$, and $S\in {\mathbb{R}}^{K\times K}$ is symmetric and positive definite, then for symmetric and positive-definite $W\in {\mathbb{R}}^{K\times K}$, $$\text{InvWishart}(W\mid \nu ,S)=\frac{1}{2^{\nu K/2}}\frac{1}{{\mathrm{\Gamma }}_K\left(\frac{\nu }{2}\right)}{\left|S\right|}^{\nu /2}{\left|W\right|}^{-(\nu +K+1)/2}\exp (-\frac{1}{2}\text{tr}(S{W}^{-1})).$$

Distribution statement

W ~ inv_wishart(nu, Sigma)

Increment target log probability density with inv_wishart_lupdf(W | nu, Sigma).

Available since 2.0

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.

Available since 2.12

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.

Available since 2.25

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.

Available since 2.0

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., $S=L_S{L}_S^{\mathrm{\top }}$ and $W=L_W{L}_W^{\mathrm{\top }}$), then the Cholesky parameterization is defined so that $$L_W\sim \text{InvWishartCholesky}(\nu ,L_S)$$ if and only if $$W\sim \text{InvWishart}(\nu ,S).$$

Probability density function

If $K\in \mathbb{N}$, $\nu \in (K-1,\mathrm{\infty })$, 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 $$\text{InvWishartCholesky}(L_W\mid \nu ,L_S)=\text{InvWishart}(L_W{L}_W^{\mathrm{\top }}\mid \nu ,L_S{L}_S^{\mathrm{\top }})\left|J_{f^{-1}}\right|,$$ where $J_{f^{-1}}$ is the Jacobian of the (inverse) transform of the variate, $f^{-1}(L_W)=L_W{L}_W^{\mathrm{\top }}$. 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 $\nu \le K-1$ 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.

Available since 2.30

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.

Available since 2.30

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.

Available since 2.30