Abstract
We construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on {mathbb {R}}^d, each containing a smoothly embedded submanifold of probability measures. The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert–Sobolev spaces and mixed-norm spaces, and support the Fisher–Rao metric as a weak Riemannian metric. Densities are expressed in terms of a deformed exponential function having linear growth. Unusually for the Sobolev context, and as a consequence of its linear growth, this “lifts” to a nonlinear superposition (Nemytskii) operator that acts continuously on a particular class of mixed-norm model spaces, and on the fixed norm space W^{2,1}; i.e. it maps each of these spaces continuously into itself. In contrast with non-parametric exponential manifolds, the density itself belongs to the model space, and the range of the chart is the whole of this space. Some of the results make essential use of a log-Sobolev embedding theorem, which also sharpens existing results concerning the regularity of statistical divergences on the manifolds. Applications to the stochastic partial differential equations of nonlinear filtering (and hence to the Fokker–Planck equation) are outlined.
Highlights
This paper constructs variants of the statistical manifolds of [25,27], for which the sample space X is Rd
This paper has developed a class of infinite-dimensional statistical manifolds that use the balanced chart of [25,27] in conjunction with a variety of probability spaces of Sobolev type
It has shown that the mixed-norm space of Sect. 4.1 is especially suited to the balanced chart, in the sense that densities belong to this space and vary continuously on the manifolds
Summary
This paper constructs variants of the statistical manifolds of [25,27], for which the sample space X is Rd. A deformed exponential function having linear growth is used in [25,27] to construct statistical manifolds modelled on the Lebesgue Lλ(μ) spaces, including the Hilbert space L2(μ) Many of these references take the classical differential geometric approach of constructing the tangent space at each point, P, in a set of measures, and building towards a global geometry. A natural direction for research in non-parametric information geometry is to adapt the manifolds outlined above to such problems by incorporating the topology of the sample space in the model space, and one way of achieving this is to use model spaces of Sobolev type This is carried out in the context of the exponential Orlicz manifold in [19], where it is applied to the spatially homogeneous Boltzmann equation.
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