Probability Distributions and Random Variables

• A discrete random variable has some finite number of outcomes drawn from a sample space:
  • E.g. the sample space for might be
  • The sample space for might be
• A probability distribution or probability mass function maps each event in the sample space to a probability between 0 and 1, where the total of all probabilities must sum to 1:
  • Disease:
    • Or, more concisely:
  • Fever:
    • Or, more concisely:

Joint Probability Distributions

The joint probability distribution represents the probability of two events, described by different random variables, happening together:

Marginalization

Conditional Probability

• Definition:
• For example:
• It follows that:
  • (the chain rule)

Independence and Conditional Independence

• Two random variables and are independent if and only if:
• and are conditionally independent given , if and only if:

Bayes Theorem / Bayes Rule

Note that

(by combining marginalization with the chain rule)

So Bayes rule can be expressed as:

Bayes Classifier

We can use Bayes rule to build a classifier:

Where corresponds to the class label and each is an attribute.

There is a serious problem with this! What is it?

Naive Bayes Classifier

• We assume that the attributes are conditionally independent given class labels, so:

• We can also recognize that is the same regardless of class so dividing with it won't change the class with the largest value.
• These leads to the naive Bayes classifier: