--- name: information-geometry-797efd6afe70 description: Fisher-Rao metric on statistical manifolds with natural gradient and divergence analysis version: 1.0.0 --- # Information Geometry Skill: Fisher-Rao Metric on Statistical Manifolds **Status**: Production Ready **Trit**: 0 (ERGODIC) **Color**: #D8E826 (Chartreuse) **Principle**: Natural gradient is parameterization-invariant optimization **Frame**: Statistical manifold with Fisher metric and dual connections --- ## Overview **Information Geometry** treats probability distributions as points on a Riemannian manifold equipped with the Fisher-Rao metric. Implements: 1. **Fisher information matrix**: g_{ij} = E[d log p / d theta_i * d log p / d theta_j] 2. **Divergences**: KL, Fisher-Rao, Hellinger, alpha-divergence, Renyi 3. **Geodesics**: m-geodesic (mixture) and e-geodesic (exponential) 4. **Natural gradient**: F^{-1} * grad (parameterization-invariant) 5. **Dually flat structure**: m-connection / e-connection pair 6. **Manifold curvature**: Scalar curvature, Amari-Chentsov tensor **Correct by construction**: Fisher-Rao is the unique Riemannian metric invariant under sufficient statistics (Chentsov's theorem). ## Core Formulae ``` Fisher information matrix: g_{ij}(theta) = E_theta[d log p(x;theta)/d theta_i * d log p(x;theta)/d theta_j] For categorical: g_{ij} = delta_{ij} / p_i (diagonal) For Gaussian: g = diag(1/sigma^2, 2/sigma^2) Fisher-Rao distance: d_FR(p,q) = 2 * arccos(sum_i sqrt(p_i * q_i)) KL divergence: KL(p||q) = sum_i p_i * log(p_i/q_i) Natural gradient: theta_new = theta - lr * F(theta)^{-1} * nabla L(theta) Dually flat structure: m-geodesic: gamma(t) = (1-t)*p + t*q (flat in mixture coords) e-geodesic: gamma(t) ~ p^{1-t} * q^t (flat in natural coords) Scalar curvature (simplex S^{n-1}): R = (n-1)(n-2)/4 ``` ## Gadgets ### 1. FisherInformation Compute Fisher information for various models: ```clojure (defn fisher-information-categorical [p] ;; g_{ij} = delta_{ij}/p_i (vec (for [i (range (count p))] (vec (for [j (range (count p))] (if (= i j) (/ 1.0 (max 1e-10 (nth p i))) 0.0)))))) (defn fisher-information-gaussian [mu sigma] [[(/ 1.0 (* sigma sigma)) 0.0] [0.0 (/ 2.0 (* sigma sigma))]]) ``` ### 2. DivergenceSuite Complete family of statistical divergences: ```clojure (kl-divergence p q) ;; asymmetric (fisher-rao-distance p q) ;; true geodesic metric (hellinger-distance p q) ;; symmetric, bounded (alpha-divergence p q alpha) ;; parametric family (renyi-divergence p q alpha) ;; order-alpha generalization ``` ### 3. NaturalGradient Parameterization-invariant optimization: ```clojure (defn natural-gradient-step [params grad fisher learning-rate] ;; theta_new = theta - lr * F^{-1} * grad (let [F-inv (matrix-inverse fisher) nat-grad (mat-vec-mul F-inv grad)] (vec-sub params (vec-scale learning-rate nat-grad)))) ``` ### 4. GeodesicTracer Trace paths on statistical manifold: ```clojure (defn mixture-connection [p q t] (mapv #(+ (* (- 1.0 t) %1) (* t %2)) p q)) (defn exponential-connection [p q t] (normalize (mapv #(* (Math/pow %1 (- 1.0 t)) (Math/pow %2 t)) p q))) ``` ## BCI Integration (Layer 18) Part of the 18-layer BCI orchestration pipeline: ### Cross-Layer Connections - **L7 Active Inference**: Free energy F = KL(Q||P) is a divergence; natural gradient minimizes it - **L17 de Rham Cohomology**: Fisher metric defines Hodge star; alpha-connections are affine connections - **L16 Spectral Methods**: Laplacian on statistical manifold via Fisher metric - **L15 Stochastic Resonance**: Fisher information maximized at resonance; SNR relates to mutual info - **L5 Riemannian Manifolds**: Fisher-Rao is a specific Riemannian metric on distribution space ### Geometry Chain: L5 -> L17 -> L18 ``` L5 (Riemannian): General curvature on signal manifold L17 (de Rham): Differential forms, Hodge theory L18 (Info Geometry): Fisher metric on probability distributions ``` --- **Skill Name**: information-geometry **Type**: Statistical Manifold / Fisher-Rao Metric / Natural Gradient **Trit**: 0 (ERGODIC) **Color**: #D8E826 (Chartreuse) **GF(3)**: Forms valid triads with PLUS + MINUS skills --- ## Integration with GF(3) Triads ``` stochastic-resonance (+1) ⊗ information-geometry (0) ⊗ derham-cohomology (-1) = 0 ✓ gay-mcp (+1) ⊗ information-geometry (0) ⊗ persistent-homology (-1) = 0 ✓ ``` --- Source attribution: plurigrid Original project: https://github.com/plurigrid/asi/blob/293ce9f76e3abcd5ba18db284688bc0d9ac6691d/skills/information-geometry/SKILL.md License: MIT License information: https://opensource.org/license/mit Original upstream instructions; downloaded from Mega AI Hub. ## Upstream license notice MIT License Copyright (c) 2025 SkillCreator.ai Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. 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IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. # Mega AI Hub resource metadata Creator / publisher: plurigrid Original project: https://github.com/plurigrid/asi/blob/293ce9f76e3abcd5ba18db284688bc0d9ac6691d/skills/information-geometry/SKILL.md Attribution basis: Source repository record Fields of work: Data & Analytics, Development & Engineering Roles: Data Analyst, Data Scientist, ML / AI Engineer Specialties: API, Backend Tools: Clojure Country applicability: Country not specified Classification basis: Source catalog labels Instruction SHA-256: b1e143b913ceb0d307e569e7076fa497872ce51c50442574a3c1d9e7a4e319bd