Bernoulli relative entropy lower bound
= Bernoulli relative entropy lower bound
{c}
{title2=$d(a,b)\geq2(a-b)^2$}
For $d(a,b)=a\log(a/b)+(1-a)\log((1-a)/(1-b))$, the <second derivative> with respect to $a$ is $1/[a(1-a)]\geq4$. Its value and first derivative at $a=b$ vanish, so $d(a,b)\geq2(a-b)^2$. Boundary values follow by continuity or the infinite-divergence convention. Together with the <binary partition bound for relative entropy>, this gives <Pinsker's inequality> with the natural-logarithm constant.