25.7 Multivariate Student-t distribution, Cholesky parameterization
25.7.1 Probability density function
Let \(K \in \mathbb{N}\), \(\nu \in \mathbb{R}^+\), \(\mu \in \mathbb{R}^K\), and \(L\) a \(K \times K\) lower-triangular matrix with strictly positive, finite diagonal then \[ \begin{array}{l} \text{MultiStudentTCholesky}(y\,\mid \nu,\,\mu,\,L) \\ = \frac{1}{\pi^{K/2}} \ \frac{1}{\nu^{K/2}} \ \frac{\Gamma\!\left((\nu + K)/2\right)} {\Gamma(\nu/2)} \ \frac{1}{\left| L \right|} \ \left( 1 + \frac{1}{\nu} \, \left(y - \mu\right)^{\top} \, L^{-T}L^{-1} \, \left(y - \mu\right) \right)^{-(\nu + K)/2} \! . \end{array} \]
25.7.2 Sampling statement
y ~ multi_student_t_cholesky(nu, mu, L)
Increment target log probability density with multi_student_t_cholesky_lupdf(y | nu, mu, L).
Available since 2.30
25.7.3 Stan functions
real multi_student_t_cholesky_lpdf(vectors y | real nu, vectors mu, matrix L)
The log of the multivariate Student-\(t\) density of vector or array of
vectors y given
degrees of freedom nu, location vector or array of vectors mu, and Cholesky factor of the scale matrix L.
For a definition of the arguments compatible with the vectors type,
see the probability vectorization section.
Available since 2.30
real multi_student_t_cholesky_lupdf(vectors y | real nu, vectors mu, matrix L)
The log of the multivariate Student-\(t\) density of vector or vector
array y given
degrees of freedom nu, location vector or vector array mu, and Cholesky factor of the scale matrix L,
dropping constant additive terms. For a definition of arguments
compatible with the vectors type, see the probability vectorization
section.
Available since 2.30
vector multi_student_t_cholesky_rng(real nu, vector mu, matrix L)
Generate a multivariate Student-\(t\) variate with degrees of freedom
nu, location mu, and Cholesky factor of the scale matrix L; may only be used in transformed data
and generated quantities blocks.
Available since 2.30
array[] vector multi_student_t_cholesky_rng(real nu, array[] vector mu, matrix L)
Generate a multivariate Student-\(t\) variate with degrees of freedom
nu, location array mu, and Cholesky factor of the scale matrix L; may only be used in transfomed data
and generated quantities blocks.
Available since 2.30
array[] vector multi_student_t_cholesky_rng(real nu, array[] row_vector mu, matrix L)
Generate an array of multivariate Student-\(t\) variate with degrees of freedom
nu, location array mu, and Cholesky factor of the scale matrix L; may only be used in transfomed data
and generated quantities blocks.
Available since 2.30