Is there a manifestly coordinate independent definition of the Fisher metric? I was reading Methods of Information Geometry by Amari and Nagaoka and Information Geometry and Its Applications by Amari, and also searched the web. All I could find was the definition with respect to the basis $\partial_{i}\log{p}$. Also, I found it odd that $X^{i}\partial_{i}\log{p}$ are interpreted as vector fields despite not being differential operators.
So, I thought about a possible treatment. What if we consider the vector fields of a statistical manifold to be just what the general definition states, with $\partial_{i} = \frac{\partial}{\partial \xi^{i}}$ as basis vectors and then define the Fisher metric to be $$g(U,V)=\int p(x) U(\log{p(x)})V(\log{p}(x))dx$$ where $U, V$ are vector fields and $\log{p(x)} : M \rightarrow R$ is a $C^{\infty}(M)$ mapping defined as $\log{p(x)}(\xi)=\log{p(x;\xi)}$. $g_{ij}$ is then the same as in the usual treatment.
Has anyone seen this approach somewhere? It seems pretty straightforward to have not been done. Also, it connects better to differential geometry as we don't have to consider $X^{i}\partial_{i}\log{p}$ as vector fields despite not satisfying the definition.