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 1ST2004 Week 7

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

3. Exam Q from 2010 ST2004 Week 73

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 74

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. IssueIs there any point changing? ST2004 Week 75

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

7. Light MetroWed 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 77

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).expert witnessMeadow'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 78

9. Brain Teasers Two regular dice are rolled. One is a 6. What’s the Pr (Other is 6) ? (Tijms, 8.1) ST2004 Week 79

10. 10 Cond Prob for Lifetimes Knowledge of current age impacts uncertainty on age at death Probability Distribution Poss LiveTimes123456 Corresp Probs0.10.20.30.30.050.05

11. ST2004 Week 711 Cond Prob for Lifetimes Knowledge of current age impacts uncertainty on age at death Expt:Choose random component Poss Lives123456 Probs0.10.20.30.30.050.05

12. Conditional Prob: Chain Rule 12ST2004 Week 7

13. Conditional Prob: Chain Rule 13ST2004 Week 7

14. Conditional Prob: Chain Rule 14ST2004 Week 7

15. Conditional Prob 15ST2004 Week 7

16. Conditional Prob 16ST2004 Week 7

17. Decomposition via Conditional Probs ST2004 Week 717

18. Decomposition via Conditional Probs ST2004 Week 718 Chance Tree

19. Decomposition via Conditional Probs ST2004 Week 719

20. Decomposition by Conditional Sim Roll regular die: note score N Toss fair coin N times What is prob no heads in N tosses? ST2004 Week 720

21. Decomposition via Conditional Probs ST2004 Week 721 Chance Tree

22. Decomposition via Conditional Probs ST2004 Week 722

23. 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. IssueIs there any point changing? ST2004 Week 723

24. 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 724

25. Decomposition by Conditional Sim ST2004 Week 725

26. Decomposition via Conditional Probs ST2004 Week 726 Chance Tree 2 1 1 2 3 3 2 3 Contestant chooses door 1, for example 3 2 2 3 Car Goat Goat Contestant does not switch; Car Goat Contestant switches Probwins = Car behind Host opens

27. Decomposition via Conditional Probs ST2004 Week 727

28. Prob Solution motivated by Conditional Sim ST2004 Week 728

29. Bayes Rule ST2004 Week 729 Inverting the Conditioning Multiple Possibilities

30. 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 730

31. 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 731 Events are T /F

32. 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 732

33. Brain Teasers ST2004 Week 733 Simulation Approach?

34. 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 734

35. Bayes Rule ST2004 Week 735 Odds Rule Form for Evidence Evidence Fusion

36. 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 736

37. 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 737

38. Serious Sally Clarke - Evidence Fusion ST2004 Week 738

39. HomeWork 2010 Exam Q Discuss Metro Tijms – Q1, Ch 122 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 739

40. Exam Q from 2010 ST2004 Week 740

41. Light MetroWed 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 741

42. 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 742

43. 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 743

44. Spam The message is reported as spam. What now is the probability that it is in fact spam? Use this calculation to illustrate Bayes Rule (i) in its conventional form (eg Q1b), (ii) in its odds-ratio form and (iii) in its Chance-Tree form. ST2004 Week 744

45. Spam You pass this through another and separate spam filter, with false positive/negative rates of 4% and 1%.It returns positive again. Suppose that the two filters to be completely independent so that errors, if any, in the two tests are independent. What now is the probability that the message is spam? ST2004 Week 745

46. Spam It is argued that the errors, if any, in the filters can’t be independent. Discuss. Does this information suggest that the probability in (c) is too large or too small? ST2004 Week 746