Search Results (3)

When using Bayesian inference, one needs to choose a prior distribution for parameters. The well-known Jeffreys prior is based on the Riemann metric tensor on a statistical manifold. Takeuchi and Amari defined the $\alpha$ -parallel prior, which generalized the Jeffreys prior by exploiting a higher-order geometric object, known as a Chentsov–Amari tensor. In this paper, we propose a new prior based on the Weyl structure on a statistical manifold. It turns out that our prior is a special case of the $\alpha$ -parallel prior with the parameter $\alpha$ equaling $-n$ , where n is the dimension of the underlying statistical manifold and the minus sign is a result of conventions used in the definition of $\alpha$ -connections. This makes the choice for the parameter $\alpha$ more canonical. We calculated the Weyl prior for univariate Gaussian and multivariate Gaussian distribution. The Weyl prior of the univariate Gaussian turns out to be the uniform prior. Full article

In this paper, we introduce a geometry called F-geometry on a statistical manifold S using an embedding F of S into the space R_X of random variables. Amari’s α-geometry is a special case of F-geometry. Then using the embedding F and a positive smooth function G, we introduce (F,G)-metric and (F,G)-connections that enable one to consider weighted Fisher information metric and weighted connections. The necessary and sufficient condition for two (F,G)-connections to be dual with respect to the (F,G)-metric is obtained. Then we show that Amari’s 0-connection is the only self dual F-connection with respect to the Fisher information metric. Invariance properties of the geometric structures are discussed, which proved that Amari’s α-connections are the only F-connections that are invariant under smooth one-to-one transformations of the random variables. Full article

Divergence functions are the non-symmetric “distance” on the manifold, Μ_θ, of parametric probability density functions over a measure space, (Χ,μ). Classical information geometry prescribes, on Μ_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 Μ_θ x Μ_θ, which induce the metric and the dual connections. Here, we construct an extension of this differential geometric structure from Μ_θ (that of parametric probability density functions) to the manifold, Μ, 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 can be interpreted as arising from an embedding of Μ_θ into Μ. 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”). Full article