Kullback's inequality

Let X_θ be a family of probability distributions on the real line indexed by the real parameter θ, and satisfying certain regularity conditions. Then $${\displaystyle \lim _{h\to 0}{\frac {D_{KL}(X_{\theta +h}\parallel X_{\theta })}{h^{2}}}\geq \lim _{h\to 0}{\frac {\Psi _{\theta }^{*}(\mu _{\theta +h})}{h^{2}}},}$$

where ${\displaystyle \Psi _{\theta }^{*}}$ is the convex conjugate of the cumulant-generating function of ${\displaystyle X_{\theta }}$ and ${\displaystyle \mu _{\theta +h}}$ is the first moment of ${\displaystyle X_{\theta +h}.}$

The left side of this inequality can be simplified as follows: $${\displaystyle {\begin{aligned}\lim _{h\to 0}{\frac {D_{KL}(X_{\theta +h}\parallel X_{\theta })}{h^{2}}}&=\lim _{h\to 0}{\frac {1}{h^{2}}}\int _{-\infty }^{\infty }\log \left({\frac {\mathrm {d} X_{\theta +h}}{\mathrm {d} X_{\theta }}}\right)\mathrm {d} X_{\theta +h}\\&=-\lim _{h\to 0}{\frac {1}{h^{2}}}\int _{-\infty }^{\infty }\log \left({\frac {\mathrm {d} X_{\theta }}{\mathrm {d} X_{\theta +h}}}\right)\mathrm {d} X_{\theta +h}\\&=-\lim _{h\to 0}{\frac {1}{h^{2}}}\int _{-\infty }^{\infty }\log \left(1-\left(1-{\frac {\mathrm {d} X_{\theta }}{\mathrm {d} X_{\theta +h}}}\right)\right)\mathrm {d} X_{\theta +h}\\&=\lim _{h\to 0}{\frac {1}{h^{2}}}\int _{-\infty }^{\infty }\left[\left(1-{\frac {\mathrm {d} X_{\theta }}{\mathrm {d} X_{\theta +h}}}\right)+{\frac {1}{2}}\left(1-{\frac {\mathrm {d} X_{\theta }}{\mathrm {d} X_{\theta +h}}}\right)^{2}+o\left(\left(1-{\frac {\mathrm {d} X_{\theta }}{\mathrm {d} X_{\theta +h}}}\right)^{2}\right)\right]\mathrm {d} X_{\theta +h}&&{\text{Taylor series for }}\log(1-t)\\&=\lim _{h\to 0}{\frac {1}{h^{2}}}\int _{-\infty }^{\infty }\left[{\frac {1}{2}}\left(1-{\frac {\mathrm {d} X_{\theta }}{\mathrm {d} X_{\theta +h}}}\right)^{2}\right]\mathrm {d} X_{\theta +h}\\&=\lim _{h\to 0}{\frac {1}{h^{2}}}\int _{-\infty }^{\infty }\left[{\frac {1}{2}}\left({\frac {\mathrm {d} X_{\theta +h}-\mathrm {d} X_{\theta }}{\mathrm {d} X_{\theta +h}}}\right)^{2}\right]\mathrm {d} X_{\theta +h}\\&={\frac {1}{2}}{\mathcal {I}}_{X}(\theta )\end{aligned}}}$$ which is half the Fisher information of the parameter θ.

The right side of the inequality can be developed as follows: $${\displaystyle \lim _{h\to 0}{\frac {\Psi _{\theta }^{*}(\mu _{\theta +h})}{h^{2}}}=\lim _{h\to 0}{\frac {1}{h^{2}}}{\sup _{t}\{\mu _{\theta +h}t-\Psi _{\theta }(t)\}}.}$$ This supremum is attained at a value of t=τ where the first derivative of the cumulant-generating function is ${\displaystyle \Psi '_{\theta }(\tau )=\mu _{\theta +h},}$ but we have ${\displaystyle \Psi '_{\theta }(0)=\mu _{\theta },}$ so that $${\displaystyle \Psi ''_{\theta }(0)={\frac {d\mu _{\theta }}{d\theta }}\lim _{h\to 0}{\frac {h}{\tau }}.}$$ Moreover, $${\displaystyle \lim _{h\to 0}{\frac {\Psi _{\theta }^{*}(\mu _{\theta +h})}{h^{2}}}={\frac {1}{2\Psi ''_{\theta }(0)}}\left({\frac {d\mu _{\theta }}{d\theta }}\right)^{2}={\frac {1}{2\operatorname {Var} (X_{\theta })}}\left({\frac {d\mu _{\theta }}{d\theta }}\right)^{2}.}$$

We have: $${\displaystyle {\frac {1}{2}}{\mathcal {I}}_{X}(\theta )\geq {\frac {1}{2\operatorname {Var} (X_{\theta })}}\left({\frac {d\mu _{\theta }}{d\theta }}\right)^{2},}$$ which can be rearranged as: $${\displaystyle \operatorname {Var} (X_{\theta })\geq {\frac {(d\mu _{\theta }/d\theta )^{2}}{{\mathcal {I}}_{X}(\theta )}}.}$$

• Kullback–Leibler divergence
• Cramér–Rao bound
• Fisher information
• Large deviations theory
• Convex conjugate
• Rate function
• Moment-generating function