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
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
Start with Kullback's inequality
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
The left side of this inequality can be simplified as follows:
which is half the Fisher information of the parameter θ.
Right side
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
We have:
which can be rearranged as:
See also
Notes and references
- ^ 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.