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

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?

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.

• (+1) Strictly speaking, it is not a closed-form solution since it still involves univariate numerical integration. However, it's better than a vanilla multivariate integration. Mar 28 at 12:52
• @utobi I totally agree. I will add a comment on that.
– New
Mar 28 at 13:31