Kullback–Leibler divergence between multivariate t and the multivariate normal?

Question

I want to calculate the Kullback–Leibler divergence between a multivariate $t$ distribution and a multivariate normal distribution, for different values of the degrees of freedom $\nu$.

However, this requires a multiple integration that seems to be difficult to calculate numerically for dimensions larger than 2. Is there a known result to calculate this integral or a numerical trick?

I understand there are general multivariate numerical integration methods. I was just wondering if there is a simpler ad hoc tool I could use as these are popular distributions, so I guess there may be some simpler tools.

@seanv507 Thanks, that is too general to be of any use, but I appreciate the pointer. — Pullback, Feb 07 '21 at 13:20
t-distribution is a scale mixture of Gaussian distributions and fining KL divergence to mixtures is notoriously difficult. You can lower bound on it if you are interested. — passerby51, Feb 07 '21 at 15:13

score 2 · Answer 1 · answered Feb 08 '21 at 08:44

There is a numerical solution based on one-dimensional numerical integrals here:

Kullback Leibler divergence between a multivariate t and a multivariate normal distributions

I doubt there is a closed form solution, but the 1D numerical integral seems simple.

Kullback–Leibler divergence between multivariate t and the multivariate normal?

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