Manifolds of differentiable densities

Newton, Nigel J.

ESAIM: Probability and Statistics, Tome 22 (2018), pp. 19-34

We develop a family of infinite-dimensional (non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class $C_b^k$ with respect to appropriate reference measures. The case $k=\infty$, in which the manifolds are modelled on Fréchet spaces, is included. The manifolds admit the Fisher-Rao metric and, unusually for the non-parametric setting, Amari’s $\alpha$-covariant derivatives for all $\alpha \in ℝ$. By construction, they are $C^{\infty }$-embedded submanifolds of particular manifolds of finite measures. The statistical manifolds are dually -embedded submanifolds of particular manifolds of finite measures. The statistical manifolds are dually (α = ±1) flat, and admit mixture and exponential representations as charts. Their curvatures with respect to the α-covariant derivatives are derived. The likelihood function associated with a finite sample is a continuous function on each of the manifolds, and the α-divergences are of class C$(\alpha =\pm 1)$ flat, and admit mixture and exponential representations as charts. Their curvatures with respect to the $\alpha$-covariant derivatives are derived. The likelihood function associated with a finite sample is a continuous function on each of the manifolds, and the $\alpha$-divergences are of class $C^{\infty }$.

Classification : 46A20, 60D05, 62B10, 62G05, 94A17
Keywords: Fisher-Rao Metric, Banach manifold, Fréchet manifold, information geometry, non-parametric statistics.

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