Entropy and Information Gain

For a classification node, entropy is computed from the proportions of its labels. With two equally common classes, using base-two logarithms gives one bit of entropy; a node containing only one class has zero. If the proportions are 75% and 25%, entropy is about 0.811 bits. This does not say one individual label is random. It summarizes the class mixture among the examples reaching that node. A candidate split divides the node into children. Calculate each child's entropy, weight it by that child's fraction of the parent examples, and subtract the weighted average from the parent's entropy. The result is information gain. Consider eight training examples, four yes and four no. Parent entropy is one bit. A split that sends three yes and one no to one child, and one yes and three no to the other, creates two children with about 0.811 bits each. They contain equal numbers of examples, so weighted child entropy is about 0.811 and information gain is about 0.189 bits. A split into perfectly pure children would yield one bit of gain in this constructed example. Decision-tree training usually compares candidate features and thresholds at each node and selects a locally attractive split. The scikit-learn tree guide describes minimizing weighted child impurity; when entropy is the criterion, that is equivalent to maximizing the reduction from the same parent node. The process is greedy, so a locally strongest split is not a guarantee of the globally best tree. A tree can keep splitting until it fits small idiosyncratic groups; depth, minimum leaf size and pruning can help control overfitting. Information gain reflects the selected impurity measure and the observed labels. It does not establish causation, explain why a feature predicts the outcome, or prove the resulting decisions are fair. Evaluate the full tree on separate data and inspect consequences for relevant groups before deployment.