Infinite-Dimensional Manifolds of Finite-Entropy Probability Measures

This paper outlines recent work by the author on infinite-dimensional statistical manifolds, employing charts that are "balanced" between mixture and exponential representations. The manifolds are distinguished from one another by the exponent of the Lebesgue spaces (Lλ (μ), 2 ≤ λ < ∞) on which they are modelled. The α-divergences have mixed second-order partial derivatives on the manifolds, enabling the construction of the Fisher metric as a pseudo-Riemannian metric. This is enough for many applications; for example, it justifies projections of Markov processes onto submanifolds in recursive estimation problems. However, higher derivatives exist when the exponent λ is 3 or more, and this leads to a limited notion of α-connections on the tangent bundle. The manifolds are also natural objects in which to embed a variety of finite-dimensional statistical manifolds. © 2013 Springer-Verlag.