Exercises

  • 1. Suppose that statistically, only 0.5 per cent of individuals are criminals. Suppose that this is the court's prior belief. Thus p(G) = 0.005. Suppose that evidence correctly identifies a guilty defendant 99 per cent of the time. That is, p(E | G) = 0.99. Suppose also that if there is no evidence, this correctly identifies an innocent defendant 99 per cent of the time. Therefore, p(NE | I) = 0.99 This means that p(E | I) = 0.01.
  • (a) Use Bayes' rule to find P(G | E), the probability that the defendant is guilty, given that evidence has been presented.
  • (b) Given that the piece of evidence has been produced, what is the probability that the defendant is innocent?
  • (c) Now suppose that evidence is not presented. What is the probability that the defendant is actually guilty, given that there is no evidence of this?
  • (d) What is the probability that he is innocent? In other words, find P(G | NE) and P (G | NE). 2 [1]

  • [1] There is a unique Nash equilibrium in the evidence-gathering game presentedin this chapter. Find this equilibrium. Show that in equilibrium, the posteriorprobability is the same as the prior, and so there is no change actual updatingof prior probabilities.
 
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