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 $\rho : X \to \mathbb{R}_+$ and a probability distribution is one such that $\sum_{x \in X} \rho(x) = 1$.

This space is actually a submanifold of $\mathbb{R}_{\geq 0}^{\vert X \vert}$. 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

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

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)