We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, $d_H^2(P,Q) = \frac{1}{{2}}\left|\left| \sqrt{P}-\sqrt{Q}\right|\right|_2^2$. That is:

\begin{equation} d_H^2(P,Q)\leq d_{TV}(P,Q). \end{equation}

I noticed, however, that when $P$ and $Q$ are Gaussian, ${\cal N}(-\mu,1)$ and ${\cal N}(\mu,1)$, $\mu\in \mathbb{R}$, truncated in the interval $[a,b]$, $a,b\in \mathbb{R}$, the following tighter bound seems to hold:

\begin{equation} d_H^2(P,Q)\leq d_H(P,Q)\leq d_{TV}(P,Q). \end{equation}

I have repeatedly verified the above inequality with numerical simulations, but I could not prove that it holds for all parameters $a,b$ and $\mu$.

If you have any pointers or ideas on how to approach this problem, I would really appreciate your input!


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.