Intro to Bayesian Learning Exercise Solutions

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Intro to Bayesian LearningExercise Solutions

• Ata Kaban
• The University of Birmingham

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• 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?
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• b) Under the conditions in the previous question,
  is it more probable that you have the disease or
  that you dont?
•  
• 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.

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• 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(B1A1). We use Bayes theorem
• Now the required quantities are known from the
  problem. These are the following
• P(A1B1)1, i.e. true positives
• P(B1)1/1000, i.e. prevalence
• P(A1B0)0.01, i.e. false positives
• P(B0)1-1/1000
• Replacing, we have

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• b) Under the conditions in the previous question,
  is it more probable that you have the disease or
  that you dont?
• ANSWER
• P(B0A1)1-P(B1A1)1-0.09099.
• So clearly it is more probable that the disease
  is not present.

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• 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
• ML maximises P(Dh) w.r.t. h, whereas MAP
  maximises P(hD). So MAP includes prior knowledge
  about the hypothesis, as P(hD) is in fact
  proportional to P(Dh)P(h). This is a good
  example where the importance and influence of
  prior knowledge is evident.
• The answer at b) is based on the maximum a
  posteriori estimate, as we have included prior
  knowledge in the form of prevalence of the
  disease. If that would not been taken into
  account, i.e. both P(B1)0.5 and P(B0)0.5 is
  considered than the hypothesis estimate would be
  the maximum likelihood one. In that case the
  presence of the disease would come out be more
  probable than the absence of it. This is an
  example of how prior knowledge can influence the
  Bayesian decisions. However, more data should be
  collected in order to produce a more reliable
  estimate.