nLab
information metric

Idea

In information geometry, a (Fisher-)information metric is a Riemannian metric on a manifold of probability distributions over some probability space $X$ (the latter often assumed to be finite).

Definition

On a finite probability space $X\in$ Set a positive measure is a function $\rho :X\to {ℝ}_{+}$ and a probability distribution is one such that ${\sum }_{x\in X}\rho (x)=1$.

This space is actually a submanifold of ${ℝ}_{\ge 0}^{\mid X\mid }$. For $\{\frac{\partial }{\partial x^i}\}$ the canonical basis of tangent vectors on this wedge of Cartesian space, the information metric $g$ is given by

$$g(\frac{\partial }{\partial x^i},\frac{\partial }{\partial x^j})(\rho )=\frac{1}{\rho (x^i)}{\delta }_{ij}.$$g(\frac{\partial}{\partial x^i}, \frac{\partial }{\partial x^j})(\rho) = \frac{1}{\rho(x^i)} \delta_{i j} \,.

Related concepts

• Wasserstein metric

References

• L. L. Campbell, An extended Čencov characterization of the information metric Journal: Proc. Amer. Math. Soc. 98 (1986), 135-141. (AMS)