Independence

Suppose there are two events:

• M: class subject is Maths.

• S: It is Sunny.

The joint probability distribution function (p.d.f.) for these events contain 4 entries.

If we want to build the joint p.d.f. we’ll have to invent those four numbers. Do we?

• We don’t have to specify with bottom level conjunctive events such as P(~M^S) IF...

• ...instead it may sometimes be more convenient for us to specify things like: P(M), P(S).

But just P(M) and P(S) don’t derive the joint distribution. So you can’t answer all questions.

“The sunshine levels do not depend on and do not influence the subject”

This can be specified very simply:

• P(S || M) = P(S) This is a powerful statement!

It required extra domain knowledge. A different kind of knowledge than numerical probabilities. It needed an understanding of causation.

From P(S || M) = P(S), the rules of probability imply:

• P(~S || M) = P(~S)

• P(M | S)=P(M)

• P(M^S)=P(M).P(S)

• P(~M ^ S) = P(~M) P(S), P(M^~S) = P(M)P(~S), P(~M^~S) = P(~M)P(~S)

A bit of notation

Assume a new event.

• R: we will do Revisions in class.

P(R || M,L) = P(R || M) P(R || ~M,L) = P(R || ~M)

“R and L are conditionally independent given M”

Given knowledge of M and S, knowing anything else in the diagram won’t help us with L, etc.

Conditional Independence formalized

A Bayes net (also called a belief network) is an augmented directed acyclic graph, represented by the pair V , E where:

• V is a set of vertices

• E is a set of directed edges joining vertices. No loops of any length are allowed.

Each vertex in V contains the following information:

• The name of a random variable.

• A probability distribution table indicating how the probability of this variable’s values depends on all possible combinations of parental values.