Infinite-Dimensional Statistical Manifolds based on a Balanced Chart

We develop a family of infinite-dimensional Banach manifolds of measures on an abstract measurable space, employing charts that are "balanced" between the density and log-density functions. The manifolds, (\tilde{M}_\lambda, \lambda\in[2,\infty)), retain many of the features of finite-dimensional information geometry; in particular, the \alpha-divergences are of class C^{\lceil \lambda \rceil-1}, enabling the definition of the Fisher metric and \alpha-derivatives of particular classes of vector fields. Manifolds of probability measures, (M_\lambda, \lambda\in[2,\infty)), based on centred versions of the balanced charts are shown to be C^{\lceil \lambda \rceil-1}-embedded submanifolds of the \tilde{M}_\lambda. The Fisher metric is a pseudo-Riemannian metric on \tilde{M}_\lambda. However, when restricted to finite-dimensional embedded submanifolds it becomes a Riemannian metric, allowing the full development of the geometry of \alpha-covariant derivatives. \tilde{M}_\lambda and M_\lambda provide natural settings for the study and comparison of a variety of embedded parametric statistical models in a large class of estimation problems.