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Chapter 7: Bayes' Rule

Chapter 7: Bayes' Rule

pp. 69-78

Authors

, University of Toronto
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Summary

One of the most useful consequences of the basic rules helps us understand how to make use of new evidence. Bayes' Rule is one key to “learning from experience.”

Chapter 5 ended with several examples of the same form: urns, shock absorbers, weightlifters. The numbers were changed a bit, but the problems in each case were identical.

For example, on page 51 there were two urns A and B, each containing a known proportion of red and green balls. An urn was picked at random. So we knew:

Pr(A) and Pr(B).

Then there was another event R, such as drawing a red ball from an urn. The probability of getting red from urn A was 0.8. The probability of getting red from urn B was 0.4. So we knew:

Pr(R/A) and Pr(R/B).

Then we asked, what is the probability that the urn drawn was A, conditional on drawing a red ball? We asked for:

Pr(A/R) =? Pr(B/R) =?

Chapter 5 solved these problems directly from the definition of conditional probability. There is an easy rule for solving problems like that. It is called Bayes' Rule.

In the urn problem we ask which of two hypotheses is true: Urn A is selected, or Urn B is selected. In general we will represent hypotheses by the letter H.

We perform an experiment or get some evidence: we draw at random and observe a red ball. In general we represent evidence by the letter E.

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