# Total variation distance bounds on multivariate normals with different means and variances

I'm trying to find a way to obtain an upper bound on the total variation distance between two multivariate normal distributions, i.e.

$$\vert \vert \mathcal{N}(\theta_1, a I) - \mathcal{N}(\theta_2, b I) \vert \vert_{TV} \le k \cdot \vert \vert \theta_1 - \theta_2 \vert \vert$$ where $$\theta_1 \ne \theta_2$$ and $$a \ne b$$ are constants and $$k$$ is some constant. Alternatively, some other bound that results in a scalar times any normed distance of $$\theta_1$$ and $$\theta_2$$ works. The problem I'm running into is I can't seem to find many ways to bound total variation distances between multivariate normal distributions. There are plenty of toy examples where either both distributions have the same mean or both distributions have the same variance in which it's easy to get an exact value, and I've computed the exact total variation distance between these two distributions, but I need something cleaner. Any help would be greatly appreciated!

## 1 Answer

This is overkill, and there may be a much simpler reference; but you may want to look at this recent paper of Devroye, Mehrabian, and Reddad [1], which handles the question in all its glorious generality and provides upper and lower bounds tight up to constants.

[1] The total variation distance between high-dimensional Gaussians, Luc Devroye, Abbas Mehrabian, and Tommy Reddad. arXiv:1810.08693, 2019.

• What does this look like for Gaussian mixtures?
– user683848
Apr 28 '20 at 19:43
• @AnnieTheKatsu I don't know -- I doubt there is a tight characterization known. Apr 28 '20 at 20:17
• by a slack characterization would also be interesting...
– user683848
Apr 28 '20 at 20:18