The Joint Distribution

Recipe for making a joint distribution of M variables:

1. Make a truth table listing all combinations of values of your variables (if there are M Boolean variables then the table will have 2M rows).

2. For each combination of values, say how probable it is.

3. If you subscribe to the axioms of probability, those numbers must sum to 1.

Using the Joint Distribution

Once you have the JD you can ask for the probability of any logical expression involving your attribute.

• P(E) = ∑ P(row)

• P(poor male) = 0.46

• P(poor) = 0.75

Inference

P(E1|E2) = (P(E1∧ E2)) / (P(E2)) = (∑ P(row)) / (∑ P(row))

P(Male | Poor) = 0.46 / 0.75 = 0.61

Joint distributions

Good news.

• Once you have a joint distribution, you can ask important questions about stuff that involves a lot of uncertainty.

Bad news.

• Impossible to create for more than about ten attributes because there are so many numbers needed when you build it.