This is an old version, view current version.

## 22.6 Multivariate Student-T Distribution

### 22.6.1 Probability Density Function

If $$K \in \mathbb{N}$$, $$\nu \in \mathbb{R}^+$$, $$\mu \in \mathbb{R}^K$$, and $$\Sigma \in \mathbb{R}^{K \times K}$$ is symmetric and positive definite, then for $$y \in \mathbb{R}^K$$, $\begin{array}{l} \text{MultiStudentT}(y\,|\,\nu,\,\mu,\,\Sigma) \\ = \frac{1}{\pi^{K/2}} \ \frac{1}{\nu^{K/2}} \ \frac{\Gamma\!\left((\nu + K)/2\right)} {\Gamma(\nu/2)} \ \frac{1}{\sqrt{\left| \Sigma \right|}} \ \left( 1 + \frac{1}{\nu} \, \left(y - \mu\right)^{\top} \, \Sigma^{-1} \, \left(y - \mu\right) \right)^{-(\nu + K)/2} \! . \end{array}$

### 22.6.2 Sampling Statement

y ~ multi_student_t(nu, mu, Sigma)

Increment target log probability density with multi_student_t_lpdf( y | nu, mu, Sigma) dropping constant additive terms.

### 22.6.3 Stan Functions

real multi_student_t_lpdf(vectors y | real nu, vectors mu, matrix Sigma)
The log of the multivariate Student-$$t$$ density of vector(s) y given degrees of freedom nu, location vector(s) mu, and scale matrix Sigma

real multi_student_t_lpdf(vectors y | real nu, row_vectors mu, matrix Sigma)
The log of the multivariate Student-$$t$$ density of vector(s) y given degrees of freedom nu, location row vector(s) mu, and scale matrix Sigma

real multi_student_t_lpdf(row_vectors y | real nu, vectors mu, matrix Sigma)
The log of the multivariate Student-$$t$$ density of row vector(s) y given degrees of freedom nu, location vector(s) mu, and scale matrix Sigma

real multi_student_t_lpdf(row_vectors y | real nu, row_vectors mu, matrix Sigma)
The log of the multivariate Student-$$t$$ density of row vector(s) y given degrees of freedom nu, location row vector(s) mu, and scale matrix Sigma

vector multi_student_t_rng(real nu, vector mu, matrix Sigma)
Generate a multivariate Student-$$t$$ variate with degrees of freedom nu, location mu, and scale matrix Sigma; may only be used in generated quantities block

vector multi_student_t_rng(real nu, row_vector mu, matrix Sigma)
Generate a multivariate Student-$$t$$ variate with degrees of freedom nu, location mu, and scale matrix Sigma; may only be used in generated quantities block

vectors multi_student_t_rng(real nu, vectors mu, matrix Sigma)
Generate an array of multivariate Student-$$t$$ variates with degrees of freedom nu, locations mu, and scale matrix Sigma; may only be used in generated quantities block

vectors multi_student_t_rng(real nu, row_vectors mu, matrix Sigma)
Generate an array of multivariate Student-$$t$$ variates with degrees of freedom nu, locations mu, and scale matrix Sigma; may only be used in generated quantities block