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For two continuous probability distributions $F,G$ and their densities, $f,g$, the (squared) Hellinger distance/affinity is given by $d^2_H(F,G)=1-\int_{\mathbb{R}} \sqrt{fg}~dx$. Suppose that $f,g$ are two-component mixture densities with mixing probability $\pi$, such that $$ f(x)=\pi f_0(x)+(1-\pi)f_1(x)\\ g(x)=\pi g_0(x)+(1-\pi)g_1(x) $$ The square root term is fairly complicated given this model. Are there any references that tackle the Hellinger distance between these mixture distributions?

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For $f=\pi f_0+(1-\pi)f_1$, $g=\pi g_0+(1-\pi)g_1$, and $\pi\in(0,1)$, we have \begin{equation} \frac{\partial^2}{\partial\pi^2}\sqrt{fg}= -\frac{\left(f_1 g_0-f_0 g_1\right){}^2}{4 (fg)^{3/2}}\le0. \end{equation} So, $\sqrt{fg}$ is concave in $\pi$ and hence $d^2_H(F,G)$ is convex in $\pi\,$: $$ d^2_H(F,G)\le\pi d^2_H(F_0,G_0)+(1-\pi)d^2_H(F_1,G_1) $$ for $\pi\in[0,1]$, where $F_j,G_j$ are the distributions with densities $f_j,g_j$, respectively, for $j=0,1$.

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