This document discusses Bayesian learning and the Bayes theorem. Some key points: - Bayesian learning uses probabilities to calculate the likelihood of hypotheses given observed data and prior probabilities. The naive Bayes classifier is an example. - The Bayes theorem provides a way to calculate the posterior probability of a hypothesis given observed training data by considering the prior probability and likelihood of the data under the hypothesis. - Bayesian methods can incorporate prior knowledge and probabilistic predictions, and classify new instances by combining predictions from multiple hypotheses weighted by their probabilities.

1. BAYESIAN
LEARNING
22, 23, 30

2. BAYES THEOREM
• Bayesian learning algorithms that calculate
explicit probabilities for hypotheses, such as
the naive Bayes classifier
• researchers show that the naive Bayes
classifier is competitive with decision tree and
neural network algorithms.

3. Features of Bayesian learning
• Each observed training example can
incrementally decrease or increase the
estimated probability that a hypothesis is
correct
• Prior knowledge can be combined with
observed data to determine the final
probability of a hypothesis

4. Features of Bayesian learning
• Bayesian methods can accommodate
hypotheses that make probabilistic
predictions
– this pneumonia patient has a 93% chance of
complete recovery")
• New instances can be classified by combining
the predictions of multiple hypotheses,
weighted by their probabilities

5. BAYES THEOREM
• determining the best hypothesis from some
space H, given the observed training data D
• the most probable hypothesis, given the data
D plus any initial knowledge
• calculate the probability of a hypothesis based
on its prior probability, the probabilities of
observing various data given the hypothesis,
and the observed data itself.

6. BAYES THEOREM
• P(h) to denote the initial probability that
hypothesis h holds
• P(h) is often called the prior probability of h
• P(D) to denote the prior probability that
training data D will be observed
• P(D/h) to denote the probability of observing
data D given in which hypothesis h holds

7. BAYES THEOREM
• Bayes theorem provides a principled way to
calculate the posterior probability of each
hypothesis given the training data

8. BAYES THEOREM
• maximally probable hypothesis is called a
maximum a posteriori (MAP) hypothesis.
• MAP hypotheses by using Bayes theorem to
calculate the posterior probability of each
candidate hypothesis.

9. Concept Learning

10. illustrate Bayes rule
• consider a medical diagnosis problem in which
there are two alternative hypotheses:
– that the patient has a particular form of cancer.
– that the patient does not
• + (positive) , - (negative)

11. illustrate Bayes rule
• We have prior knowledge that over the entire
population of people only .008 have this
disease.
• a correct positive result in only 98% of the
cases in which the disease is actually present
and a correct negative result in only 97% of
the cases in which the disease is not present.

12. illustrate Bayes rule
• we now observe a new patient for whom the
lab test returns a positive result
• Should we diagnose the patient as having
cancer or not?
• posteriori hypothesis can be found using