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Let us define a distance metric between two joint probability math functions $p(x,y)$ and $q(x,y)$ as in the following

\begin{align} \sum_{y}\sqrt{\sum_{x}p(x)\left(p(y|x)-q(y|x)\right)^2}. \end{align}

By Jensen inequality, we have

\begin{align*} \sum_{y}\sqrt{\sum_{x}p(x)\left(p(y|x)-q(y|x)\right)^2}&\geq\sum_{x,y}\lvert p(x,y)-q(x,y)\rvert\\ &=\lVert p(x,y)-q(x,y)\rVert_1 \end{align*}

where $\lVert\cdot\rVert_1$ is the total variation distance.

What is the name of this distance? Is there any other relationship between this distance and any other stochastic distances?

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  • $\begingroup$ are you assuming $q(x,y) = q(y|x)p(x)$? $\endgroup$ – Memming Sep 11 '17 at 23:43
  • $\begingroup$ some sort of nested Csiszár's f-divergence? $\endgroup$ – Memming Sep 11 '17 at 23:50
  • $\begingroup$ Is this symmetric? $\endgroup$ – Liviu Nicolaescu Sep 12 '17 at 1:42
  • $\begingroup$ It is symmetric for $p(x)=q(x)$. $\endgroup$ – Math_Y Sep 12 '17 at 3:29
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\begin{align*} \sum_{y}\sqrt{\sum_{x}p(x)\left(p(y|x)-p(y)\right)^2} &=\sum_{y}p(y)\sqrt{\sum_{x}p(x)\left(\frac{p(y|x)}{p(y)}-1\right)^2}\\ &=\sum_{y}p(y)\sqrt{\sum_{x}p(x)\left(\frac{p(x|y)}{p(x)}-1\right)^2}\\ &=\sum_{y}p(y)\sqrt{\sum_{x}\frac{p^2(x|y)}{p(x)}-2p(x|y)+p(x)}\\ &=\sum_{y}p(y)\sqrt{\sum_{x}\frac{p^2(x|y)}{p(x)}-1}\\ &=\mathbb{E}_Y\sqrt{\chi^2\left(p(x|y)\parallel p(x)\right)} \end{align*} where $\chi^2(\cdot||\cdot)$ is chi-square distance.

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