Fisher–Rao metric

E837390

Riemannian metric information geometric structure statistical distance

The Fisher–Rao metric is a fundamental Riemannian metric on statistical manifolds that quantifies the intrinsic geometric structure of families of probability distributions via the Fisher information.

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Statements (46)

Predicate	Object
instanceOf	Riemannian metric ⓘ information geometric structure ⓘ statistical distance ⓘ
alternativeName	Fisher information metric NERFINISHED ⓘ
appliesTo	exponential families ⓘ parametric families of probability distributions ⓘ
associatedWith	C. R. Rao NERFINISHED ⓘ Ronald A. Fisher NERFINISHED ⓘ
basedOn	Fisher information NERFINISHED ⓘ
characterizedBy	uniqueness as the monotone Riemannian metric on classical statistical models ⓘ
componentExpression	expected outer product of score functions ⓘ negative expected Hessian of log-likelihood under regularity conditions ⓘ
definedOn	manifold of probability distributions ⓘ statistical manifold ⓘ
determines	Riemannian volume element on statistical manifold ⓘ
domain	interior of parameter space where Fisher information is finite ⓘ
field	differential geometry ⓘ information geometry ⓘ statistics ⓘ
gives	Riemannian metric tensor on parameter space of a statistical model ⓘ
hasProperty	positive definite (for identifiable models) ⓘ symmetric bilinear form on tangent spaces of statistical manifold ⓘ
induces	Levi-Civita connection on statistical manifold NERFINISHED ⓘ geodesics on space of probability distributions ⓘ
is	canonical Riemannian metric on a statistical manifold ⓘ invariant under reparametrization of the statistical model ⓘ invariant under sufficient statistics ⓘ monotone under Markov morphisms in classical statistics ⓘ
localApproximationOf	Kullback–Leibler divergence NERFINISHED ⓘ
quantifies	intrinsic geometric structure of families of probability distributions ⓘ
relatedTo	Fisher information matrix NERFINISHED ⓘ Jeffreys prior (via volume element) NERFINISHED ⓘ Kullback–Leibler divergence (locally) NERFINISHED ⓘ natural gradient in optimization ⓘ
specialCaseOf	monotone metrics in quantum information (classical limit) ⓘ
underlies	natural gradient descent ⓘ
usedFor	Cramér–Rao lower bound interpretation ⓘ asymptotic theory of estimation ⓘ defining geodesic distance between probability distributions ⓘ measuring infinitesimal distinguishability of probability distributions ⓘ studying curvature of statistical models ⓘ
usedIn	Bayesian statistics NERFINISHED ⓘ information theory ⓘ machine learning ⓘ signal processing ⓘ statistical physics ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Hellinger distance → relatedTo → Fisher–Rao metric ⓘ