Is the Fisher information matrix an Einstein like tensor?

Question

I read a bit https://en.m.wikipedia.org/wiki/Fisher_information and the fact that the Fisher information matrix is a riemannian metric and the curvature of the relative entropy makes me wonder if we can view this matrix as a tensor which would play with regard to relative entropy the same role as Einstein's tensor with regard to the stress-energy tensor in Einstein's field equation. Are there works pointing toward such an analogy? Can we generalize the stress-energy tensor to a stress-energy-entropy tensor that would be the source of a "gravitational-informational" field described by a similar equation taking the form $F_{ij}=kS_{ij}$ with $F_{ij}$ the Fisher information and $S_{ij}=(p_{i}\log p_{j})$ an entropy matrix/tensor for its informational part?