PPE Midterm

PPE Midterm - 1. Bayes theorem: (a) What is the purpose of...

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1. Bayes theorem: (a) What is the purpose of Bayesian analysis? (b) What is the relationship between prior probabilities and posterior probabilities? The purpose of Bayes’s theorm is to determine what the probability is that some hypothesis is true given that we have observed some data. Bayes’s theorem is especially useful since we do not have perfect tests. Therefore, the data will always be flawed in some way – perfect information does not exist in reality, as much as we may think we can approach the limit of the truth. So how do we get to the probability that the test is representative of reality? As an example, let us consider a medical test in which the test is not 100% accurate. A rational doctor would ask him or herself, given that I have a positive test result for sickness, what is the probability that the person is actually sick? If we say that p(H) is the prior probability, then p(D|H) is the posterior probability. If we look at p(D|H), this is will help us calculate the probability of health given some certain datum, or rather, given the probability of a positive test result, what is the probability that the test is true. Here, the doctor is given some prior probability of health for this patient (e.g. this person’s age, weight, sex, etc means a 30% chance of sickness). Using this information, the doctor can observe some data (D) to calculate the posterior probability p(D|H) to determine the appropriate decision to be made in regards to this patient Human’s are not good at calculating the prior data The form of Bayes’s theorem is: p(H|D) = p(D|H)p(H) / {[p(D|H)p(H)] + [p(D| not-H)p(not-H)]} - - prior probabilities to calculate posterior probability o p(H) is the prior probability o p(D|H) is the posterior probability - - Medical test, given sickness is not 100% o Given that you have a positive test result for sickness, what is the probability that the person is actually sick p(H|D) – probability of health given the data – given the probability of a positive test result, what is the probability that the test is true p(H|D) = p(D+H) / [p(D+H) + p(D+ not-H)] p(H|D) = p(D|H)p(H) / {[p(D|H)p(H)] + [p(D| not-H)p(not-H)]} use this for the homework Bayes’s Theorem d = data h = hypothesis
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p(D) = p(not-H + D) + p(H + D) P(not-H + D) = p(D | not-H) x p(not-H) P(H + D) = p(D | H) x p(H) Thus p(H | D) = p(H + D) / [p(not-H + D) + p(H + D)] = P(D | H) x p(H) P(H| D) = [p(D|H) x p(H)] / [[P(D|H)x(p(H)] + [p(D | not-H) x p(not-H)]] There is a strong relationship between Bayes’s theorem and conditional assessment Ratio form of Bayes’s theorem p(D | H) × p(H) p(H | D) = --------------------------------------------- [ p(D | H) × p(H) + p(D | not-H) × p(not-H) ] p(D | not-H) × p(not-H) p(not-H | D) = --------------------------------------------- [ p(D | H) × p(H) + p(D | not-H) × p(not-H) ] p(H | D) p(D | H) × p(H) p(D | H) p(H) ------------ = ----------------------- = ------------ × -------- p(not-H | D) p(D | not-H) × p(not-H)
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PPE Midterm - 1. Bayes theorem: (a) What is the purpose of...

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