## 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