Axiomatic Geometry of Conditional Models

We formulate and prove an axiomatic characterization of the Riemannian geometry underlying manifolds of conditional models. The characterization holds for both normalized and nonnormalized conditional models. In the normalized case, the characterization extends the derivation of the Fisher information by Cencov while in the nonnormalized case it extends Campbell's theorem. Due to the close connection between the conditional I-divergence and the product Fisher information metric, we provides a new axiomatic interpretation of the geometries underlying logistic regression and AdaBoost.