# Kullback-Leibler divergence between multivariate t and the multivariate normal?

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.

• Monte Carlo integration? – seanv507 Feb 7 at 12:17
• @seanv507 Thanks, that is too general to be of any use, but I appreciate the pointer. – Pullback Feb 7 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 7 at 15:13

## 1 Answer

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.