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In information theory and statistics, Kullback's inequality is a lower bound on the Kullback–Leibler divergence expressed in terms of the large deviations rate function. If P and Q are probability distributions on the real line, such that P is absolutely continuous with respect to Q, i.e. P<<Q, and whose first moments exist, then

$D_{KL}(P\Vert Q)\ge {\Psi }_Q^{\ast }(\mu {\text{'}}_1(P)),$

where ${\Psi }_Q^{\ast }$ is the rate function, i.e. the convex conjugate of the cumulant-generating function, of $Q$, and $\mu {\text{'}}_1(P)$ is the first moment of $P.$

The Cramér–Rao bound is a corollary of this result.

Proof

Let P and Q be probability distributions (measures) on the real line, whose first moments exist, and such that P<<Q. Consider the natural exponential family of Q given by

$Q_{\theta }(A)=\frac{{\int }_A{e}^{\theta x}Q(dx)}{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\theta x}Q(dx)}=\frac{1}{M_Q(\theta )}{\int }_A{e}^{\theta x}Q(dx)$

for every measurable set A, where $M_Q$ is the moment-generating function of Q. (Note that Q₀=Q.) Then

$D_{KL}(P\Vert Q)=D_{KL}(P\Vert Q_{\theta })+{\int }_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}P}(\log \frac{\mathrm{d}{Q}_{\theta }}{\mathrm{d}Q})\mathrm{d}P.$

By Gibbs' inequality we have $D_{KL}(P\Vert Q_{\theta })\ge 0$ so that

$D_{KL}(P\Vert Q)\ge {\int }_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}P}(\log \frac{\mathrm{d}{Q}_{\theta }}{\mathrm{d}Q})\mathrm{d}P={\int }_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}P}(\log \frac{e^{\theta x}}{M_Q(\theta )})P(dx)$

Simplifying the right side, we have, for every real θ where $M_Q(\theta )<\mathrm{\infty }:$

$D_{KL}(P\Vert Q)\ge \mu {\text{'}}_1(P)\theta -{\Psi }_Q(\theta ),$

where $\mu {\text{'}}_1(P)$ is the first moment, or mean, of P, and ${\Psi }_Q=\log M_Q$ is called the cumulant-generating function. Taking the supremum completes the process of convex conjugation and yields the rate function:

$D_{KL}(P\Vert Q)\ge \underset{\theta }{sup}\{\mu {\text{'}}_1(P)\theta -{\Psi }_Q(\theta )\}={\Psi }_Q^{\ast }(\mu {\text{'}}_1(P)).$

Corollary: the Cramér–Rao bound

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Start with 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

$\underset{h\to 0}{lim}\frac{D_{KL}(X_{\theta +h}\Vert X_{\theta })}{h^2}\ge \underset{h\to 0}{lim}\frac{{\Psi }_{\theta }^{\ast }({\mu }_{\theta +h})}{h^2},$

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

Left side

The left side of this inequality can be simplified as follows:

$\underset{h\to 0}{lim}\frac{D_{KL}(X_{\theta +h}\Vert X_{\theta })}{h^2}=\underset{h\to 0}{lim}\frac{1}{h^2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(\log \frac{\mathrm{d}{X}_{\theta +h}}{\mathrm{d}{X}_{\theta }})\mathrm{d}{X}_{\theta +h}$

$=\underset{h\to 0}{lim}\frac{1}{h^2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}[(1-\frac{\mathrm{d}{X}_{\theta }}{\mathrm{d}{X}_{\theta +h}})+\frac{1}{2}{(1-\frac{\mathrm{d}{X}_{\theta }}{\mathrm{d}{X}_{\theta +h}})}^2+o\left({(1-\frac{\mathrm{d}{X}_{\theta }}{\mathrm{d}{X}_{\theta +h}})}^2\right)]\mathrm{d}{X}_{\theta +h},$

where we have expanded the logarithm $\log x$ in a Taylor series in $1-1/x$,

$=\underset{h\to 0}{lim}\frac{1}{h^2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[\frac{1}{2}{(1-\frac{\mathrm{d}{X}_{\theta }}{\mathrm{d}{X}_{\theta +h}})}^2\right]\mathrm{d}{X}_{\theta +h}$

$=\underset{h\to 0}{lim}\frac{1}{h^2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[\frac{1}{2}{\left(\frac{\mathrm{d}{X}_{\theta +h}-\mathrm{d}{X}_{\theta }}{\mathrm{d}{X}_{\theta +h}}\right)}^2\right]\mathrm{d}{X}_{\theta +h}=\frac{1}{2}{ℐ}_X(\theta ),$

which is half the Fisher information of the parameter θ.

Right side

The right side of the inequality can be developed as follows:

$\underset{h\to 0}{lim}\frac{{\Psi }_{\theta }^{\ast }({\mu }_{\theta +h})}{h^2}=\underset{h\to 0}{lim}\frac{1}{h^2}\underset{t}{sup}\{{\mu }_{\theta +h}t-{\Psi }_{\theta }(t)\}.$

This supremum is attained at a value of t=τ where the first derivative of the cumulant-generating function is $\Psi {\text{'}}_{\theta }(\tau )={\mu }_{\theta +h},$ but we have $\Psi {\text{'}}_{\theta }(0)={\mu }_{\theta },$ so that

${\Psi }^{\prime }{\text{'}}_{\theta }(0)=\frac{d{\mu }_{\theta }}{d\theta }\underset{h\to 0}{lim}\frac{h}{\tau }.$

Moreover,

$\underset{h\to 0}{lim}\frac{{\Psi }_{\theta }^{\ast }({\mu }_{\theta +h})}{h^2}=\frac{1}{2{\Psi }^{\prime }{\text{'}}_{\theta }(0)}{\left(\frac{d{\mu }_{\theta }}{d\theta }\right)}^2=\frac{1}{2\mathrm{V}\mathrm{a}\mathrm{r}(X_{\theta })}{\left(\frac{d{\mu }_{\theta }}{d\theta }\right)}^2.$

Putting both sides back together

We have:

$\frac{1}{2}{ℐ}_X(\theta )\ge \frac{1}{2\mathrm{V}\mathrm{a}\mathrm{r}(X_{\theta })}{\left(\frac{d{\mu }_{\theta }}{d\theta }\right)}^2,$

which can be rearranged as:

$\mathrm{V}\mathrm{a}\mathrm{r}(X_{\theta })\ge \frac{(d{\mu }_{\theta }/d\theta {)}^2}{{ℐ}_X(\theta )}.$

See also

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