Independence of Attributes

This is actually quite interesting and maybe very useful for my Poker AI.

It’s this idea that if you have two attributes, how do you know they are independent?

• For example, having a Modern Vaccine and a Placebo Vaccine, how do you know the Moderna vaccine is better?
• (iffy) Having a Poker AI play against a human, how do you know the Poker AI is better than them?

The idea is that we can do Hypothesis Testing to check if data is independent, where

• $H_0$: Attributes are independent
• $H_1$: Attributes are not independent

Steps

Step 1: Calculate Expected Table $e_{ij}=\frac{(i^{th}\text{ row total})(j^{th}\text{ row total})}{\text{total}}$

• You can derive the formula above, actually it’s more intuitive to calculate as $total\ast P(row)\ast P(col)$, which simplifies to the equation above Step 2: Calculate $\lambda$

$\lambda =2\sum \sum y_{ij}\ln \frac{y_{ij}}{e_{ij}}\sim {\chi }^2((a-1)(b-1))$ where

• $y_{ij}$ = observed
• $e_{ij}$ = expected
• a = # rows
• b = # columns

Step 3: Calculate p-value, where $\Lambda \sim {\chi }^2((a-1)(b-1))$

We use the p-value to come to conclusions about $H_0$ and $H_1$ (see p-value to come to conclusions about $H_0$ and $H_1$ (see Hypothesis Testing).

Equality of Proportions

Testing for independence of attributes is the same as testing the equality of proportions. The proportions should be the same if they are independent.