7
$\begingroup$

I want to calculate the Kullback–Leibler divergence

enter image description here

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,

enter image description here

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

$\endgroup$

1 Answer 1

6
$\begingroup$

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.

enter image description here

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.

$\endgroup$
2
  • 1
    $\begingroup$ (+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. $\endgroup$
    – utobi
    Mar 28 at 12:52
  • 1
    $\begingroup$ @utobi I totally agree. I will add a comment on that. $\endgroup$
    – New
    Mar 28 at 13:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.