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Reasoning with Probs

Reasoning with Probs. How does evidence lead to conclusions in situations of uncertainty? Bayes Theorem

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Reasoning with Probs

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  1. Reasoning with Probs How does evidence lead to conclusions in situations of uncertainty? Bayes Theorem Data fusion, use of techniques that combine data from multiple sources and gather that information in order to achieve inferences, which will be more efficient and potentially more accurate than if they were achieved by means of a single source. Spam Cancer Screening Law Exams ST2004 Week 7

  2. Probability RulesConditional Prob and Independence Multiplication Rule All computed probs: depend on real world knowledge assumptions about real world Sometimes Useful to be explicit What - if ST2004 Week 7

  3. Exam Q from 2010 ST2004 Week 7

  4. Brain Teasers • Two regular dice are rolled. One is a 6. What’s the Pr (Other is 6) ? (Tijms, 8.1) • Who is the murderer? (Tijms, Ch 1 Q6)? Murder committed; know either X or Y – equally likely. Evidence: actual perp has blood group A 10% of people group A; X is group A Seek Pr( X is perp | evidence) ST2004 Week 7

  5. Brain Teasers • Monty Hall Game Show (Tijms, Ch 1, Q11) One car, behind one of three doors. Player selects one: say Door 1 Before opening this door host opens one of two others: say Door 2 GOAT! host offers chance to change selection. Issue Is there any point changing? ST2004 Week 7

  6. Life Expectancy in Ireland Average age at death, given survival to 60, = 79 Average age at death = 75 Average age at death = 80 http://understandinguncertainty.org/node/272 ST2004 Week 7

  7. Light Metro Wed 10 Nov 2010 Teens at risk from hyper-texting Teenagers who send more than 100 text messages per day are more likely to have had sex, tried drugs, research has revealed. 4200 students at 20 schools; hyper-texting 19.2% Such teens 43% more likely to have tried alcohol. ST2004 Week 7

  8. Serious Sally Clarke - Sudden Infant Death SID The case was widely criticised because of the way statistical evidence was misrepresented in the original trial, particularly by Meadow. He stated in evidence as an expert witness that "one sudden infant death in a family is a tragedy, two is suspicious and three is murder unless proven otherwise" (Meadow's law). He claimed that, for an affluent non-smoking family like the Clarks, the probability of a single cot death was 1 in 8,543, so the probability of two cot deaths in the same family was around "1 in 73 million" (8543 × 8543). ST2004 Week 7

  9. CondProb for LifetimesKnowledge of current age impacts uncertainty on age at death Probability Distribution PossLiveTimes 1 2 3 4 5 6 CorrespProbs 0.1 0.2 0.3 0.3 0.05 0.05 ST2004 Week 7

  10. Conditional Prob: Chain Rule ST2004 Week 7

  11. Conditional Prob: Chain Rule ST2004 Week 7

  12. Decomposition via Conditional Probs ST2004 Week 7

  13. Decomposition via Conditional Probs Chance Tree ST2004 Week 7

  14. Decomposition via Conditional Probs Chance Tree ST2004 Week 7

  15. Brain Teasers Monty Hall Game Show (Tijms, Ch 1, Q11) One car, behind one of three doors. Player selects one: say Door 1 Before opening this door host opens one of two others: say Door 2 GOAT! host offers chance to change selection. Issue Is there any point changing? ST2004 Week 7

  16. Decomposition by Conditional Sim Contestant always chooses Door 1 Car behind random door – equal probs Host actions Don’t Switch Do Switch Prize Car behind 1 2 3 Pseudo Code ST2004 Week 7

  17. Decomposition by Conditional Sim ST2004 Week 7

  18. Contestant Contestant switches does not switch; 2 3 G oat Car Contestant chooses 1 3 2 door Goat Car 1, for example 3 2 Car Goat 2 1 Goat 3 2 3 Car Prob wins = Decomposition via Conditional Probs Chance Tree Car behind Host opens ST2004 Week 7

  19. Bayes Rule Inverting the Conditioning Multiple Possibilities ST2004 Week 7

  20. Brain Teasers • Who is the murderer? (Tijms, Ch 1 Q6)? Murder committed; know either X or Y – equally likely. Evidence: actual perp has blood group A 10% of people group A; X is group A Seek Pr( X is perp | evidence) ST2004 Week 7

  21. Brain Teaser Roll regular die: note score N Toss fair coin N times Observe no heads. What now Pr(scored 2)? Pr(scored 2|no heads)? ST2004 Week 7

  22. Bayes Rule Odds Rule Form for Evidence Evidence Fusion ST2004 Week 7

  23. Serious Sally Clarke - Sudden Infant Death SID  He claimed that, for an affluent non-smoking family like the Clarks, the probability of a single cot death was 1 in 8,543, so the probability of two cot deaths in the same family was around "1 in 73 million" (8543 × 8543). ST2004 Week 7

  24. HomeWork • 2010 Exam Q • Discuss Metro • Tijms • Q1, Ch 1 22 players + ref Friend bets €10 at least one common birthday What is fair price of the bet • Q12, Ch 1 Told family has two children; one is a daughter Prob other is daughter? Told family has two children; ring bell – girl opens Prob other also a girl? ST2004 Week 7

  25. Spam Exam Q A simple spam filter is used on a single incoming message. You know only a 1% chance of such messages are spam. You also know that the filter is imperfect - with false positive (ie positive for spam given not spam) and false negative rates of 5% and 2% respectively. Defining events F± and S in a natural way, restate this information in terms ST2004 Week 7

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