Question

Proof that if the same attribute appears twice in a decision tree, than the information gain equals zero when you apply it for the second time. You should assume that it is possible that other attributes are tested between the two applications of this attribute.

Answer 1

The best way to test this is to play video games which include taking quick choices or decisions. Imagine you are a leader fighting for freedom. You have the option of taking the violent route or peaceful route. The 1st time you take this decision your brain takes a while to think of the outcomes of each situation. Through each level of struggle you will have to make the decision of either being peaceful or violent. The 2nd time you take the decision you already know the kind of outcomes that will arise. Therefore almost reflexly you choose the same path the 2nd time as you are familiar with it. Its almost like how you take your hand away from a flame without much thought. After all reaction is the reflex of thought.