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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.

Arya McCarthy
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Pullback
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  • Monte Carlo integration? – seanv507 Feb 07 '21 at 12:17
  • @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
  • This paper has what you are looking for. The final solution is on page 8 and is quite cumbersome. https://link.springer.com/content/pdf/10.1007/s40304-019-00200-8.pdf – Snowy Baboon Oct 31 '23 at 01:08

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

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

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

Jean Pierre
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