# Non-centered parameterization of the multivariate student-t distribution

In the multivariate normal case, we have

``````mu + L * z
``````

where `mu` is the mean vector, `L` is the cholesky factor of the covariance matrix and `z` are independent standard normal variables. My question is how to correctly incorporate the degrees of freedom parameter `nu` into the above formula to create multivariate-t random variables? I remember having seens this somewhere already, but I can’t find it right now…

One may optimize it using the inv_chi_square and turn the division in a multiplication.

3 Likes

Thanks, that’s what I was looking for! :-)

The parameterization: mu+L*z you mentioned for the multivariate normal is valid when the covariance matrix is known (defined in the data block).

Is there any difference when the covariance matrix is unknown (defined in the parameters block)?

Thanks a lot.

It’s still valid.

I think it’s a bit more complicated in my case. I’ll create a new topic.

Could somebody re-post the solution at the end of this thread?

The link to the google group seems to no longer be valid.

``````data {
int<lower=1> p;
vector[p] mu;
cholesky_factor_cov[p,p] L;
real<lower=0> nu;
}
parameters {
vector[p] z;
real<lower=0> u;
}
transformed parameters {
vector[p] x;
x = mu + sqrt(nu / u) * (L * z); // distributed multi_student_t
}
model {
target += normal_lpdf(z | 0, 1);
target += chi_square_lpdf(u | nu);
}``````
3 Likes