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28.4 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 LS and LW (i.e., S=LSLS and W=LWLW), then the Cholesky parameterization is defined so that LWInvWishartCholesky(ν,LS) if and only if WInvWishart(ν,S).

28.4.1 Probability density function

If KN, ν(K1,), and LS,LWRK×K are lower triangular matrixes with positive diagonal elements, then the Cholesky parameterized inverse Wishart density is InvWishartCholesky(LWν,LS)=InvWishart(LWLWν,LSLS)|Jf1|, where Jf1 is the Jacobian of the (inverse) transform of the variate, f1(LW)=LWLW. The log absolute determinant is log|Jf1|=Klog(2)Kk=1(Kk+1)logLWk,k.

The probability functions will raise errors if νK1 or if LS and LW are not Cholesky factors (square, lower-triangular matrices with positive diagonal elements) of the same size.

28.4.2 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