# Intro to Bayesian Learning Exercise Solutions - PowerPoint PPT Presentation

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Intro to Bayesian Learning Exercise Solutions. Ata Kaban The University of Birmingham.

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Intro to Bayesian Learning Exercise Solutions

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## Intro to Bayesian LearningExercise Solutions

Ata Kaban

The University of Birmingham

You are to be tested for a disease that has prevalence in the population of 1 in 1000. The lab test used is not always perfect: It has a false-positive rate of 1%. [A false-positive result is when the test is positive, although the disease is not present.] The false negative rate of the test is zero. [A false negative is when the test result is negative while in fact the disease is present.]

• a) If you are tested and you get a positive result, what is the probability that you actually have the disease?

• b) Under the conditions in the previous question, is it more probable that you have the disease or that you don’t?

• c) Would the answers to a) and / or b) differ if you use a maximum likelihood versus a maximum a posteriori hypothesis estimation method? Comment on your answer.

ANSWER a) We have two binary variables, A and B. A is the outcome of the test, B is the presence/absence of the disease. We need to compute P(B=1|A=1). We use Bayes theorem:

Now the required quantities are known from the problem. These are the following:

• P(A=1|B=1)=1, i.e. true positives

• P(B=1)=1/1000, i.e. prevalence

• P(A=1|B=0)=0.01, i.e. false positives

• P(B=0)=1-1/1000

Replacing, we have:

b) Under the conditions in the previous question, is it more probable that you have the disease or that you don’t?