Parametrized measure models

Abstract

We develop a new and general notion of parametric measure models and statistical models on an arbitrary sample space $\Omega$ which does not assume that all measures of the model have the same null sets. This is given by a differentiable map from the parameter manifold $M$ into the set of finite measures or probability measures on $\Omega$, respectively, which is differentiable when regarded as a map into the Banach space of all signed measures on $\Omega$. Furthermore, we also give a rigorous definition of roots of measures and give a natural characterization of the Fisher metric and the Amari–Chentsov tensor as the pullback of tensors defined on the space of roots of measures. We show that many features such as the preservation of this tensor under sufficient statistics and the monotonicity formula hold even in this very general set-up.