Entropy & Information Gain
Entropy counts how surprising a node's labels are, in bits. Information gain is how much a candidate question removes that surprise — the score that picks every split in ID3/C4.5 trees — plus the gain-ratio fix for the trap where high-cardinality columns "win" by memorizing.
01.The Problem: Which Question Deserves to Be First?
You are building a decision tree (Topic 56: a flowchart of if-then questions that predicts by which box the data lands in).
At the root, the tree must pick one question to split on.
Your data has 40 candidate columns.
"Weather?" "Age?" "ID number?" — what makes one question better than another?
"Better" means: after asking it, the piles it creates should be less mixed.
If you split 100 customers by "visited site last week", you might get one pile where everyone buys and one where nobody does. Perfectly sorted.
If you split by a useless column, both piles stay 50/50. Still hopelessly mixed.
So you need a number that measures messiness. One number. Per pile. Comparable across questions.
Claude Shannon gave us that number in 1948. It is called entropy.
From Impurity to Split Choice 🌲
From Impurity to Split Choice 🌲
Entropy measures disorder in bits; information gain measures how much a split removes it; gain ratio corrects the bias that favors high-cardinality attributes.
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