What is the relationship between the Fisher Information and the Fisher Information metric? It seems that there are two separate definitions for the Fisher information, and I'm wondering if there is some relationship between the two.
The first is the so-called Fisher information which appears in some versions of the log-Sobolev inequality. It has the form $I(f) = \int_X \frac{|\nabla f|^2}{f} dx$. Notice that the derivatives are in the observation space $X$. This quantity seems to be important in functional analysis.
The second is the so called Fisher information metric or Fisher-Rao metric. For a parameterized family of probability densities $f(x,\theta)$, we can express it in the following way:
$$g^{FR}_f(\theta_i, \theta_j) = \int_X \frac{\nabla_{\theta_i} f \,\nabla_{\theta_j} f}{f} dx.$$
Note that the derivatives here are in the statistical manifold, not the observation space. It's straightforward to generalize this to a non-parametrized model with Frechet derivatives. This metric is important in statistics and probability because it is in some sense a canonical metric.
What I'm trying to understand is whether there some relationship between these two. I can see that if $X$ is the real line, and we perturb $f$ by translating it, then the $I(f)$ and the norm of the translation in $g_f^{FR}$ are the same. However, I don't see much else relating them.
Does anyone have any pointers?
 A: This is not a complete answer, but a comment that's gotten too long plus a hint.
My hunch is that it is unlikely that there is a general, simple connection. My reasoning is the following: 
Let $X$ be a smooth manifold. The Fisher-Rao metric is invariant under the action of diffeomorphisms by pushforward, while the Fisher information functional $I$ is not; even under scalings on $X=\mathbb{R}^n$, the squared norm of the gradient will pick up a non-trivial scaling factor.
But maybe there is hope to find an interesting relation when the statistical manifold is closed under isometries, as you hint at with your translation example.

ADDENDUM: I found the following paper: B. Khesin, G. Misiolek, and K. Modin, “Geometric hydrodynamics via Madelung transform,” Proc. Natl. Acad. Sci. U.S.A., vol. 115, no. 24, pp. 6165–6170, Jun. 2018.
In Proposition 13 there, the Fisher metric and Fisher functional interact to produce some sort of wave equation that seems to be of interest. I'm afraid I cannot say much more. Maybe someone else will read this and be able to expand on this direction.
