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The First Ruler of Distinguishability

Why Fisher information is the local geometry of statistical models

information geometry

fisher information

pullback metric

riemannian metric

This post introduces Fisher information as the first “ruler” for probability distributions: a local metric that measures how distinguishable nearby statistical models are from data. We will see why it arises naturally as a pullback metric, why Chentsov’s theorem makes it unique, and how it foreshadows the deeper connection between KL divergence, natural gradients, and learning.

Jun 5, 2026

Gaurav Khanal

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The First Ruler of Distinguishability