Connection between Fisher metric and the relative entropy
Can someone prove the following connection between Fisher information metric and the relative entropy (or KL divergence) in a purely mathematical rigorous way?
$$D( p(cdot , a+da) | p(.,a) ) =frac{1}{2} g_{i,j} da^{i} da^{j} + (O( |da|^{3})$$
where $a=(a^1,dots, a^n), da=(da^1,dots,da^n)$, $$g_{i,j}=int partial_i (log p(x;a)) partial_j(log p(x;a))~ p(x;a)~dx$$ and $g_{i,j} da^{i} da^{j}:=sum_{i,j}g_{i,j} da^{i} da^{j}$ 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.
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 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 prior distributions 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 have a rigorous proof, you’ll have 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 densities $f$ and $g$ is defined as
$$
D[f,g] = int (f(x) – g(x)) logleft(frac{f(x)}{g(x)} right) dx , .
$$
If we have a family of densities parameterized by $theta=(theta_1,dots,theta_k)$, then
$$
D[p(,cdot,midtheta), p(,cdot,midtheta + Deltatheta)] = int ( p(x,midtheta) – p(xmidtheta + Deltatheta)) logleft( frac{p(xmidtheta)}{p(xmidtheta + Deltatheta)}right) ,dx , ,
$$
where $Deltatheta=(Deltatheta_1,dots,Deltatheta_k)$. Introducing the notation
$$
Delta p(xmidtheta) = p(xmidtheta) – p(xmidtheta + Deltatheta) , ,
$$
some simple algebra gives
$$
D[p(;cdot,midtheta), p(;cdot,midtheta + Deltatheta)] = intfrac{Delta p(xmidtheta)}{p(xmidtheta)} logleft(1+frac{Delta p(xmidtheta)}{p(xmidtheta)}right)p(xmidtheta),dx , .
$$
Using the Taylor expansion for the natural logarithm, we have
$$
logleft(1+frac{Delta p(xmidtheta)}{p(xmidtheta)}right) approx frac{Delta p(xmidtheta)}{p(xmidtheta)} , ,
$$
and therefore
$$
D[p(;cdot,midtheta), p(;cdot,midtheta + Deltatheta)] approx intleft(frac{Delta p(xmidtheta)}{p(xmidtheta)}right)^2p(xmidtheta),dx , .
$$
But
$$
frac{Delta p(xmidtheta)}{p(xmidtheta)} approx frac{1}{p(xmidtheta)} sum_{i=1}^k frac{partial p(xmidtheta)}{partialtheta_i} Deltatheta_i = sum_{i=1}^k frac{partial log p(xmidtheta)}{partialtheta_i} Deltatheta_i , .
$$
Hence
$$
D[p(,cdot,midtheta), p(,cdot,midtheta + Deltatheta)] approx sum_{i,j=1}^k g_{ij} ,Deltatheta_i Deltatheta_j , ,
$$
where
$$
g_{ij} = int frac{partial log p(xmidtheta)}{partialtheta_i} frac{partial log p(xmidtheta)}{partialtheta_j} p(xmidtheta) ,dx , .
$$
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.
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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