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Kullback's inequality

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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.[1] 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 where is the rate function, i.e. the convex conjugate of the cumulant-generating function, of , and is the first moment of

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

Proof

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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 for every measurable set A, where is the moment-generating function of Q. (Note that Q0 = Q.) Then By Gibbs' inequality we have so that Simplifying the right side, we have, for every real θ where where is the first moment, or mean, of P, and is called the cumulant-generating function. Taking the supremum completes the process of convex conjugation and yields the rate function:

Corollary: the Cramér–Rao bound

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Start with Kullback's inequality

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Let Xθ be a family of probability distributions on the real line indexed by the real parameter θ, and satisfying certain regularity conditions. Then

where is the convex conjugate of the cumulant-generating function of and is the first moment of

Left side

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The left side of this inequality can be simplified as follows: which is half the Fisher information of the parameter θ.

Right side

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The right side of the inequality can be developed as follows: This supremum is attained at a value of t=τ where the first derivative of the cumulant-generating function is but we have so that Moreover,

Putting both sides back together

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We have: which can be rearranged as:

See also

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Notes and references

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  1. ^ Fuchs, Aimé; Letta, Giorgio (1970). "L'inégalité de Kullback. Application à la théorie de l'estimation". Séminaire de Probabilités de Strasbourg. Séminaire de probabilités. 4. Strasbourg: 108–131.