Multi student-t with df per dimension

specification

#1

The multi_student_t function has a nu argument accepting a real value that influences how heavy-tailed the distribution is, but I’d like to be able to accommodate multivariate data that has a different degree of heavy-tailed-ness in each dimension. I vaguely recall someone already solved this in a prior post but I’m having trouble finding it now. Any suggestions?


#2

The basic idea is to use univariate Student t marginal distributions with different numbers of degrees of freedom and a multivariate copula to capture the dependence, which is discussed in
https://scholar.google.com/scholar?cluster=1395612595769919176&hl=en&as_sdt=0,33


#3

I actually found the thread, which turned out to be over in the old google group:
https://groups.google.com/d/msg/stan-users/nk03n8hX4Us/-pTN5IGjCQAJ

You indicate there that “you don’t really need to utilize a copula for this but copulas would be interesting for other reasons”; care to elaborate?


#4

Oops, I guess that thread was regarding the regular ol’ multi_student_t, but I followed up here on the different df’s per dimension case.


#5

Ah, I now remember where I got stuck last time I looked at this. The code worked out in the linked google thread (and below) achieved expressing a model parameter as mult_student_t-with-different-nu-per-dimension, but I’m having trouble working out how to express a data variable as thus distributed. Any suggestions?

For posterity, here’s the code from the google thread:

data {
    int<lower=1> p; //number of dimensions
    vector[p] mu; //mean
    cholesky_factor_cov[p,p] L; //cholesky factor of the covariance
    vector<lower=0>[p] nu; //nu per dimension
}
parameters {
    vector[p] z; //helper variable, to be normal(0,1)
    vector<lower=0>[p] u; //helper variable, to be chi_square(nu)
}
transformed parameters {
    vector[p] x;
    // implies x ~ multi_student_t_2
    x = mu + sqrt(nu / u) * (L * z); 
}
model {
    z ~ normal(0, 1);
    u ~ chi_square(nu);
}

#6

You should have what you need to make this happen.

I think there’s something funky with sqrt(nu / u), cause if that’s a vector and L * z is a vector, I don’t see how they multiply together. Assuming that sqrt(nu / u) stuff can get crammed in L, call that \hat{L}.

Anyway, so you can evaluate p(z), but what you have is x and a 1-to-1 transformation between x and z. In terms of the multivariate change of variables in the manual (pg. 403 of 2.17) that is:

x = \mu + \hat{L} z = f(z)
p_x(x) = p_z(f^{-1}(x)) | \text{det} J_{f^{-1}}(x)|

Then

z = \hat{L}^{-1}(x - \mu) = f^{-1}(x)
J_{f^{-1}}(x) = \hat{L}^{-1}
\text{det} \hat{L}^{-1} = \frac{1}{\text{det} \hat{L}}

And the determinant of a lower triangular matrix is just the product of the diagonal elements.


#7

x = mu + sqrt(nu ./ u) .* (L * z);


#8

I’m interested also in multi student t with df per dimension and found this through search. Copulas are new to me. However, reading the past linked google group thread - am I correct in saying the solution posted above stops short of using a copula ?