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Fisher Information & Information Geometry

Fisher gives the natural metric on probability distributions—not Euclidean distance in parameters

Concept 55 of 100TheoryPhase 10
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#55Fisher InfoTheory
key equation
Fij(θ)=E[∂ilog⁡pθ⋅∂jlog⁡pθ]F_{ij}(\theta) = \mathbb{E}\left[\partial_i \log p_\theta \cdot \partial_j \log p_\theta\right]
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Concept 55 of 100Fisher InfoTheory / Phase 10: Mathematical foundations & information geometry
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Distance in parameter space should mean "distinguishability of distributions"—Fisher captures this

Fij(θ)=E[∂ilog⁡pθ⋅∂jlog⁡pθ]F_{ij}(\theta) = \mathbb{E}\left[\partial_i \log p_\theta \cdot \partial_j \log p_\theta\right]
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Why It Matters for Modern Models

  • Fisher gives the natural metric on probability distributions—not Euclidean distance in parameters
  • Explains why KL penalties in RLHF/PPO are geometric constraints, not arbitrary regularization
  • Connects curvature to uncertainty: high Fisher = parameters are well-identified

What Tutorials Skip

What is still poorly explained in textbooks and papers:

  • Distance in parameter space should mean "distinguishability of distributions"—Fisher captures this
  • The Cramér-Rao bound: variance of any estimator ≥ 1/Fisher—more info = tighter estimates
  • Fisher is the Hessian of KL at θ=θ₀, making it a second-order object without needing the loss Hessian

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Key Equation
Fij(θ)=E[∂ilog⁡pθ⋅∂jlog⁡pθ]F_{ij}(\theta) = \mathbb{E}\left[\partial_i \log p_\theta \cdot \partial_j \log p_\theta\right]

The Fisher Information Matrix measures how distinguishable distributions are:

Fij(θ)=Ex∼pθ[∂log⁡pθ(x)∂θi∂log⁡pθ(x)∂θj]F_{ij}(\theta) = \mathbb{E}_{x \sim p_\theta}\left[\frac{\partial \log p_\theta(x)}{\partial \theta_i} \frac{\partial \log p_\theta(x)}{\partial \theta_j}\right]

Equivalently (under regularity):

Fij(θ)=−Ex∼pθ[∂2log⁡pθ(x)∂θi∂θj]F_{ij}(\theta) = -\mathbb{E}_{x \sim p_\theta}\left[\frac{\partial^2 \log p_\theta(x)}{\partial \theta_i \partial \theta_j}\right]

KL as local metric: For small parameter changes:

KL(pθ∥pθ+dθ)≈12dθ⊤F(θ)dθ\text{KL}(p_\theta \| p_{\theta + d\theta}) \approx \frac{1}{2} d\theta^\top F(\theta) d\theta

Canonical Papers

Information Geometry and Its Applications

Amari2016Springer
Read paper →

Natural Gradient Works Efficiently in Learning

Amari1998Neural Computation
Read paper →

Connections

Prerequisites

ℒML/CE/KL∀Theory

Enables

🧭Natural Grad🪞Bregman

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