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Bayes Rule

Bayes Rule. N total people: This is the much bigger blue box including G and g. Everyone with trait G. The basic idea here: no one cares about “G” except as it affects the probability that the person is “g”. Everyone with trait g. G. g G. g. Prior odds ratio of a random

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Bayes Rule

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  1. Bayes Rule N total people: This is the much bigger blue box including G and g Everyone with trait G The basic idea here: no one cares about “G” except as it affects the probability that the person is “g” Everyone with trait g G g G g Prior odds ratio of a random person being a spy: from Scotchmer 1998 Statistical Reasoning in Court

  2. Statistical Reasoning in Court: Collins • LA mugging. Yellow car. Interracial couple, she with a blond ponytail, he with a moustache. Court case: The defendants match. What should we make of this? Convict them? • N=couples in population; Priors Prior odds of guilt: Posterior odds of guilt? Posterior to what? Answer, the match. E.g., the pink area. from Scotchmer 1998 Statistical Reasoning in Court

  3. Statistical Reasoning: More on Collins Suppose the only evidence is the match to the description. Suppose there are 2 couples in LA that match: G, the pink circle Posterior probability of (g)uilt: Posterior odds ratio, (g)uilt to (i)nnocence: Preponderance of evidence: odds ratio is larger than one Beyond a reasonable doubt: odds ratio is very large from Scotchmer 1998 Statistical Reasoning in Court

  4. Statistical ReasoningRed buses and blue buses • City has one bus line but two bus companies. Pedestrian is run over; no witnesses. There are 4 times as many blue buses as red buses. • Prior probabilities of guilt and innocence: (g) = 4/5 and (i)=1/5 • With no evidence, the posterior probability is the same as the prior probability. The probability ratio of guilt to innocence is 4/1. By the preponderance-of-evidence standard, trier of fact should convict. • Victim’s family sues the blue-bus company on grounds that it is 4-to-1 likely that the perpetrator was a blue bus. • Should the court hold the company liable? from Scotchmer 1998 Statistical Reasoning in Court

  5. Statistical ReasoningMore on red buses and blue buses Hm…. No one likes this outcome!! In the court case that this is based on, the court disallowed such a conclusion. Observations: • The whole point of liability is to deter negligence. But if the blue company is always convicted, there is no deterrence of negligence by the red company. • If the blue bus company is always convicted, then the blue drivers will be more careful, leading to the conclusion that they do not provide four times as many accidents. Thus the premise is wrong. The statistics must account for equilibrium behavior. • Is it fair to convict the bus company without identifying the driver? Homework: What is the odds ratio for a particular blue-bus driver? from Scotchmer 1998 Statistical Reasoning in Court

  6. Statistical reasoning: Probabilistic Effects • 1950’s, nuclear testing in Nevada. • Later, cancer appeared (leukemia). • Epidemiological data: (these numbers are slightly wrong) The leukemia rate is 3 cases per 1000 people. In Nevada the rate is 6 cases per 1000 people. • For an individual case, is the AEC liable? There is no specific evidence that the AEC caused the victim’s leukemia. The prior probability that the AEC is guilty is (g)=1/2. The posterior is the same, so the probability ratio is (g)/ (i)=1, which just meets the preponderance of evidence standard. • Problem: No standard of evidence will assign liability correctly. The AEC is liable either for all the cancers or for none of them. from Scotchmer 1998 Statistical Reasoning in Court

  7. Statistical Reasoning: Order Statistics • Cancer Clusters: Woburn has a leather-tanning plant and a chemical plant, and it turns out that the town has a leukemia cluster. • The town has five times the ordinary cancer rate. • Should the plants be liable? • Some town will be the highest order statistic. What do we make of this? from Scotchmer 1998 Statistical Reasoning in Court

  8. Apply to evidence techniques • Fingerprinting? What does a match mean? What are the dark blue, light blue and pink areas? • DNA match • Repressed memory? Is this in the same category? from Scotchmer 1998 Statistical Reasoning in Court

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