Data Mining - Entropy (Information Gain)

Entropy is a function “Information” that satisfies:

where:

• p1p2 is the probability of event 1 and event 2
• p1 is the probability of an event 1
• p1 is the probability of an event 2

where:

• H stands for entropy
• E for Ensemble ???

The entropy of a distribution with finite domain is maximized when all points have equal probability.

Bigger is the entropy, more is the event unpredicatble

Higher entropy means there is more unpredictability in the events being measured .

Higher entropy mean that the events being measured are less predictable.

100% predictability = 0 entropy

Fifty Fifty is an entropy of one.

Why is picking the attribute with the most information gain beneficial? It reduces entropy, which increases predictability. Information gain is positive when there is a decrease in entropy from choosing classifier/representation. A decrease in entropy signifies an decrease in unpredictability, which also means an increase in predictability.

3 - Example

3.1 - Flipping a Coin

A two class problem.

3.2 - Rolling a die

A six class problem.

$$\begin{array}{rrl} Entropy & = & - \sum_{x}{ p_x log_2( p_x)} \\ & = & - 6 * (\frac{1}{6} log_2 (\frac{1}{6})) \\ & \approx & 2.58 \end{array}$$

3.3 - Rolling a weighted die

The weighted die is more predictable than a fair die.

4 - How unpredictable is your data?

4.1 - Titanic

Titanic training set with a two class problem: survived or die

4.1.1 - 342 survivors

case: 342 survivors on a total of 891 passengers

4.1.2 - 50 survivors

case: 50 survivors on a total of 891 passengers

It's a more predictable data set.

5 - Document / Reference

• Bill Howe (UW)