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# Bayes - ECON*2740ClassNotes,Winter2012...

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1 ECON*2740 Class Notes, Winter 2012 Course Instructor: David Prescott Topics: Bayes’ Theorem & and Introduction to Discrete Random Variables Bayes’ Theorem Bayes’ theorem provides a way to update probabilities in the light of new information. In the following example, new information is provided by a drug seeking dog. Suppose that at Toronto airport, 1 in a thousand packages originating from a certain country contains drugs. Suppose a drug hunting dog correctly identifies 99% of packages that contain drugs and correctly identifies 98% of packages that do not contain drugs. What is the probability that a package identified as containing drugs actually does contain drugs? What is the probability that a package identified as not containing drugs actually contains drugs? D: a package contains drugs ܦ : a package does not contain drugs Y: the dog indicates a package contains drugs N: the dog indicates a package does not contain drugs Prior to information provided by the dog, P(D) = 0.001 The dog’s accuracy is described by the following conditional probabilities ܲሺܻ|ܦሻ ൌ 0.99 ܽ݊݀ ܲሺܰ|ܦ ሻ ൌ 0.98 Bayes’ theorem can be used to calculate the required probabilities. Recall the definition of conditional probability:

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2 ܲሺܣ|ܤሻ ൌ ܲሺܣ ת ܤሻ ܲሺܤሻ This can be rewritten as: ܲሺܣ ת ܤሻ ൌ ܲሺܣ|ܤሻܲሺܤሻ Similarly, ܲሺܣ ת ܤሻ ൌ ܲሺܤ|ܣሻܲሺܣሻ Hence, it follows that ܲሺܣ|ܤሻܲሺܤሻ ൌ ܲሺܤ|ܣሻܲሺܣሻ From which we get Bayes’ theorem: ሾ1ሿ ܲሺܣ|ܤሻ ൌ ܲሺܤ|ܣሻܲሺܣሻ ܲሺܤሻ In many interesting problems, Bayes’ theorem can be used in the form shown in [1]. In other cases, such as the drug detection problem described above it is useful to express ܲሺܤሻ ൌ ܲሺܤ ת ܣሻ ൅ ܲሺܤ ת ܣ ҧ ܲሺܤሻ ൌ ܲሺܤ|ܣሻܲሺܣሻ ൅ ܲሺܤ|ܣ ҧ ሻܲሺܣ ҧ This and [1] give a second form of Bayes’ theorem: ሾ2ሿ ܲሺܣ|ܤሻ ൌ ܲሺܤ|ܣሻܲሺܣሻ ܲሺܤ|ܣሻܲሺܣሻ ൅ ܲሺܤ|ܣ ҧ ሻܲሺܣ ҧ Let’s apply [2] to the first question: What is the probability that a package identified as containing drugs actually does contain drugs?
3 ሾ3ሿ ܲሺܦ|ܻሻ ൌ ܲሺܻ|ܦሻܲሺܦሻ ܲሺܻ|ܦሻܲሺܦሻ ൅ ܲሺܻ|ܦ ሻܲሺܦ From the information provided: P(D) = 0.001, P(Y|D) = 0.99, and since ܲሺܰ|ܦ ሻ ൌ 0.98 it follows that ܲሺܻ|ܦ ሻ ൌ 0.02 . ܲሺܦ|ܻሻ ൌ 0.99ݔ0.001 0.99ݔ0.001 ൅ 0.02ݔ0.999 0.00099 0.02097 ൌ 0.0472 Despite the dog’s low error rates (1% false negatives and 2% false positives) it turns out that less than 5% of packages the dog identifies as containing drugs actually do contain drugs. The dog’s “evidence” turns out to be quite weak evidence of a crime. More than 95% of the “accused” are actually innocent.

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Bayes - ECON*2740ClassNotes,Winter2012...

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