Information gain is a frequently used metric in calculating the gain during a split in tree-based methods.
First o all, the entropy of a dataset if defined as
$$ S = - sum_i p_i \log p_i - sum_i (1-p_i)\log p_i, $$
where $p_i$ is the probability of a class.
The information gain is the difference between the entropy.
For example, in a decision tree algorithm, we would split a node. Before splitting, we assign a label $m$ to the node,
$$ S_m = - p_m \log p_m - (1-p_m)\log p_m. $$
After the splitting, we have two groups that contributes to the entropy, group $L$ and group $R$,
$$ S’_m = p_L (- p_m \log p_m - (1-p_m)\log p_m) + p_R (- p_m \log p_m - (1-p_m)\log p_m), $$
where $p_L$ and $p_R$ are the probabilities of the two groups. Suppose we have 100 samples before splitting and 29 samples in the left group and 71 samples in the right group, we have $p_L = 29/100$ and $p_R = 71/100$.
The information gain is thus
$$ Gain = S_m - S’_m. $$
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