# Relationship between KL, chi-squared, and Hellinger

There are many well-known relationships between the KL divergence, chi-squared ($$\chi^2$$) divergence, and the Hellinger metric. In the paper "Assouad, Fano, and Le Cam" by Bin Yu, the author appears to use the following relationship (see Example 2, p. 433): $$\text{KL}(f,g) \le \int \frac{(\sqrt{f}-\sqrt{g})^2}{f}\,dx.$$ This looks almost like the standard relationship $$\text{KL}(f,g)\le \chi^2(f,g)$$, but differs in the square roots taken in the numerator. I have not seen this general relationship previously.

My question is: Does this inequality hold for general densities $$f,g$$?

(Note: In the paper, this inequality is applied to special choices of $$f$$ and $$g$$, and so maybe the author only means it for these functions. It is not clear to me based on the writing.)

This inequality is false in general.

E.g., let $$f$$ and $$g$$ be pdf's on $$[0,1]$$ given by the formulas $$f=1$$ and $$g=f+t\,1_{(0,1/2)}-t\,1_{(1/2.1)}$$ for $$t\in(0,1)$$. Then the left- and right-hand sides of your inequality are, respectively, $$\sim t^2/2$$ and $$\sim t^2/4$$ as $$t\downarrow0$$, so that your inequality fails to hold.

Moreover, if here $$t\uparrow1$$, then the left- and right-hand sides of your inequality go, respectively, to $$\infty$$ and $$1-1/\sqrt2$$. So, your inequality will fail to hold even if any extra universal constant factor is inserted on its right-hand side.

Here is the graph of the ratio the left-hand side of your inequality to its right-hand side:

The reason for such counterexamples is that $$\text{KL}(f,g)=\int f\ln\frac fg$$ can be arbitrarily large if $$g$$ is allowed to take values much smaller than the corresponding values of $$f$$, whereas the right-hand side of your inequality will remain bounded if $$f$$ is (say) bounded away from $$0$$ on the set where $$g>0$$.

• Tried to build on this nice counterexample by giving a counterexample that is piecewise constant and another which is smooth. I think this shows the inequality does not hold even under the conditions cited in Section 29.3 of the paper shared by the OP. Aug 10 at 18:35
• @NawafBou-Rabee : Thank you for your comment. Aug 10 at 19:03

The inequality does not hold for even simple densities which fulfill the conditions given in Section 29.3 of the paper shared by the OP.

Here is a piecewise constant counterexample, $$f(x) = \frac{3}{2} 1_{\{(0,1/2)\}}(x) + \frac{1}{2} 1_{\{ (1/2,1) \}}(x)$$ $$g(x) = \frac{1}{2} 1_{\{(0,1/2)\}}(x) + \frac{3}{2} 1_{\{ (1/2,1) \}}(x)$$ These functions are (i) probabilities densities whose support is $$(0,1)$$; (ii) they are bounded from above/below; and (iii) piecewise constant. However, the KL divergence between the corresponding distributions is $$\int_0^1 f(x) \log(f(x)/g(x)) dx = \log(3)/2 \ge 1/2$$ whereas the claimed upper bound is $$\int_0^1 \frac{(\sqrt{f(x)} - \sqrt{g(x)})^2}{f(x)} dx = \frac{(\sqrt{6}-\sqrt{2})^2}{3} \le 1/2 \;.$$

And, here is a smooth counterexample, $$f \equiv 1$$ $$g(x) = 2 (1-x)$$ In this case, by direct integration, the KL divergence is $$(2-4\sqrt{2})/3 > 1/4$$ while the claimed upper bound is $$2 - 4 \sqrt{2}/3<1/4$$.

All this seems to point to a possible bug in Example 2 of the paper.