Vector student-t reparameterisation for fat tails

Hello! This question relates to section 24.7 in the manual on reparameterising the student-t distribution based on the product of a normal and gamma distribution to avoid fat tails (24.7 Reparameterization | Stan User’s Guide).

Instead of a single beta, as in the manual, suppose instead I have a vector beta.

parameters {
  real<lower=0> nu;
  real<lower=0> tau;
  vector[n] alpha;
  // ...
transformed parameters {
  vector[n] beta;
  beta = alpha / sqrt(tau);
  // ...

Do I use a single, common tau for all calculations. Or do I also need a vector[n] tau?


I think you would need a vector of tau. If your joint density is

p(\beta_1, \beta_2) = \mathrm{Student(\beta_1 | \nu, 0, 1)} \mathrm{Student(\beta_2 | \nu, 0, 1)}

then each of those get expanded out separately.

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Great - thank you!