Nonparametric Information Geometry: From Divergence Function to Referential-Representational Biduality on Statistical Manifolds

Abstract

Divergence functions are the non-symmetric "distance" on the manifold, M θ , of parametric probability density functions over a measure space, (X, µ). Classical information geometry prescribes, on M θ : (i) a Riemannian metric given by the Fisher information; (ii) a pair of dual connections (giving rise to the family of α-connections) that preserve the metric under parallel transport by their joint actions; and (iii) a family of divergence functions (α-divergence) defined on M θ × M θ , which induce the metric and the dual connections. Here, we construct an extension of this differential geometric structure from M θ (that of parametric probability density functions) to the manifold, M, of non-parametric functions on X, removing the positivity and normalization constraints. The generalized Fisher information and α-connections on M are induced by an α-parameterized family of divergence functions, reflecting the fundamental convex inequality associated with any smooth and strictly convex function. The infinite-dimensional manifold, M, has zero curvature for all these α-connections; hence, the generally non-zero curvature of M θ can be interpreted as arising from an embedding of M θ into M. Furthermore, when a parametric model (after a monotonic scaling) forms an affine submanifold, its natural and expectation parameters form biorthogonal coordinates, and such a submanifold is dually flat for α = ±1, generalizing the results of Amari's α-embedding. The present analysis illuminates two different types of duality in information geometry, one concerning the referential status of a point (measurable function) expressed in the divergence function ("referential duality") and the other concerning its representation under an arbitrary monotone scaling ("representational duality").

Duality of Maximum Entropy and Minimum Divergence

“…However, there are many related studies. For example, a nonparametric information geometry on the space of all functions without constraints for positivity and normalizing is discussed in Zhang [40]. Amari [41] characterizes (ρ, τ )-divergence with decomposable dually flat structure, see also [42].…”

Geometry Induced by a Generalization of Rényi Divergence

“…Clearly, expression (21) implies that ∂g ∂κ (α, κ) > 0 for all α ∈ (0, α 0 ) and κ > 0. We proved that items (i)-(iii) are satisfied.…”

“…As consequence, the derivative of κ(α) exists at any α ∈ (α 0 , α 1 ). Expression (16) for the derivative of κ(α) follows from (17), (20) and (21).…”

“…The ϕ-divergence is intrinsically related to the (ρ, τ)-model of Zhang, which was proposed in [19,20], extended to the infinite-dimension setting in [21], and explained in more details in [22,23].…”

On Hölder Projective Divergences

“…This work is further extended in [14] where Zhang stresses the two different types of duality in information geometry: the referential duality and the representational duality (with the study of the (ρ, τ)-geometry for monotone embeddings).…”