# References for Hellinger distance/affinity involving mixture distributions

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?

## 1 Answer

For $$f=\pi f_0+(1-\pi)f_1$$, $$g=\pi g_0+(1-\pi)g_1$$, and $$\pi\in(0,1)$$, we have $$$$\frac{\partial^2}{\partial\pi^2}\sqrt{fg}= -\frac{\left(f_1 g_0-f_0 g_1\right){}^2}{4 (fg)^{3/2}}\le0.$$$$ So, $$\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$$.