## 25.6 Multivariate Student-t distribution

### 25.6.1 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 $$y \in \mathbb{R}^K$$, $\begin{array}{l} \text{MultiStudentT}(y\,|\,\nu,\,\mu,\,\Sigma) \\ = \frac{1}{\pi^{K/2}} \ \frac{1}{\nu^{K/2}} \ \frac{\Gamma\!\left((\nu + K)/2\right)} {\Gamma(\nu/2)} \ \frac{1}{\sqrt{\left| \Sigma \right|}} \ \left( 1 + \frac{1}{\nu} \, \left(y - \mu\right)^{\top} \, \Sigma^{-1} \, \left(y - \mu\right) \right)^{-(\nu + K)/2} \! . \end{array}$

### 25.6.2 Sampling statement

y ~ multi_student_t(nu, mu, Sigma)

Increment target log probability density with multi_student_t_lupdf(y | nu, mu, Sigma).
Available since 2.0

### 25.6.3 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
Available since 2.18

real multi_student_t_lupdf(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 dropping constant additive terms
Available since 2.25

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
Available since 2.18

real multi_student_t_lupdf(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 dropping constant additive terms
Available since 2.25

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
Available since 2.18

real multi_student_t_lupdf(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 dropping constant additive terms
Available since 2.25

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
Available since 2.18

real multi_student_t_lupdf(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 dropping constant additive terms
Available since 2.25

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 transformed data and generated quantities blocks
Available since 2.0

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 transfomed data and generated quantities blocks
Available since 2.18

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 transformed data and generated quantities blocks
Available since 2.18

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 transformed data andgenerated quantities blocks
Available since 2.18