Dapatkah seseorang membuktikan hubungan berikut antara metrik informasi Fisher dan entropi relatif (atau divergensi KL) dengan cara yang benar-benar ketat secara matematis?
where , and is the Einstein summation convention.
I found the above in the nice blog of John Baez where Vasileios Anagnostopoulos says about that in the comments.
Jawaban:
In 1946, geophysicist and Bayesian statistician Harold Jeffreys introduced what we today call the Kullback-Leibler divergence, and discovered that for two distributions that are "infinitely close" (let's hope that Math SE guys don't see this ;-) we can write their Kullback-Leibler divergence as a quadratic form whose coefficients are given by the elements of the Fisher information matrix. He interpreted this quadratic form as the element of length of a Riemannian manifold, with the Fisher information playing the role of the Riemannian metric. From this geometrization of the statistical model, he derived his Jeffreys's prior as the measure naturally induced by the Riemannian metric, and this measure can be interpreted as an intrinsically uniform distribution on the manifold, although, in general, it is not a finite measure.
To write a rigorous proof, you'll need to spot out all the regularity conditions and take care of the order of the error terms in the Taylor expansions. Here is a brief sketch of the argument.
The symmetrized Kullback-Leibler divergence between two densitiesf and g is defined as
If we have a family of densities parameterized byθ=(θ1,…,θk) , then
This is the original paper:
Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems. Proc. Royal Soc. of London, Series A, 186, 453–461.
sumber
Proof for usual (non-symmetric) KL divergence
Zen's answer uses the symmetrized KL divergence, but the result holds for the usual form as well, since it becomes symmetric for infinitesimally close distributions.
Here's a proof for discrete distributions parameterized by a scalarθ (because I'm lazy), but can be easily re-written for continuous distributions or a vector of parameters:
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You can find a similar relationship (for a one-dimensional parameter) in equation (3) of the following paper
The authors refer to
for a proof of this result.
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