Information geometry is the Riemannian geometry of statistical models. A parametrised family of probability distributions is treated as a manifold whose points are the distributions and whose coordinates are the parameters. The canonical metric is the Fisher information
Property. A dual-flat pair of connections
Beyond the metric, the family carries a one-parameter family of
-connections, whose two extremes are flat and dual to each other. For an exponential family these are the exponential and mixture connections, denoted and . The natural parameter and the expectation parameter are the two affine coordinate systems, related by the Legendre transform of the log-partition . The Fisher metric is . Whether this whole -package is the image of a categorical construction is an open question.
Example. Divergence as canonical geometry
The Kullback–Leibler divergence is the Bregman divergence of
, and its second order is the Fisher metric. Projecting a distribution onto a family is a Bregman/Pythagorean projection along a dual-flat geodesic. This is the geometry in which MBAR-style estimators and flow-matching paths on the simplex are naturally read.