Kullback–Leibler divergence between two multivariate t distributions with different degrees of freedom?

Question

I want to calculate the Kullback–Leibler divergence

between two multivariate $t$ distributions with different degrees of freedom (say $\nu_1$ and $\nu_2$), but same location and scale matrix, for arbitrary dimensions,

How can I deal with the multivariate integral in the KL divergence? Is there a closed-form solution?

Answer 1

Check the following post:

Kullback Leibler divergence between two multivariate t distributions

The authors reduce the dimension of the integral from $p$ (in your notation) to 1, which seems to be easier to handle. The post contains numerical examples in R.

Edit. As mentioned by @utobi in a comment, it is not a closed-form solution since it still involves univariate numerical integration. However, it's better than a vanilla multivariate integration. Indeed, this solution is a more "numerically tractable" alternative to multivariate integration, but not a closed-form one.