Gain ratio for a categorical feature

The problem, from first principles

Information gain has a flaw: a feature with many distinct values (an ID column, say) splits the data into tiny, perfectly pure groups and looks wonderful, while telling you nothing about unseen samples. Gain ratio divides the gain by how finely the feature splits the data, so many-valued features are penalised.

From theory to code

Implement gain_ratio(feature, labels). Split labels into one group per distinct value of feature, compute the information gain of that split, then divide by the split's intrinsic value. Use base-2 logarithms throughout.

Constraints

• feature and labels are one-dimensional arrays of equal length; either can hold arbitrary integers.
• Return a Python float.
• Return 0.0 for empty input, and 0.0 when the feature has a single distinct value (its intrinsic value is zero).
• Use base-2 logs, so entropy is measured in bits.

Hints

Open one at a time. Each gives away a little more than the last.