Thinking with Probability

Transcription

1 Thinking with Probability Thinking fast and slow Thinking by decomposing events formalising probability rules conditional probability Brain Teasers Tijms Problems, Chap 1 2 dice rolled; one is 6. What prob other is 6? Aged 22 Prob of death Monty Hall; Choose 1; Learn goat at 2. Pr(car at 3)? 2 babies dead. Pr(mother murderer)? ST Week 6 1

2 Thinking with Probability Thinking about systems by decomposing events Formalising probability rules conditional probability Forward modelling A makes B more likely Inverse thinking Have observed B; Pr(A B) Evaluating evidence Bayes Rule Using evidence in prediction Statistical Inference ST Week 6 2

3 Evaluating Evidence 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.. ST Week 6 3

4 All Probs are Conditional Probability Measure of uncertainty about event given info Pr(A wins league, given probs for each match, and indep) Pr(A wins league, given that somebody wins, and probs, and indep) Pr(A wins league, given no info, other than probs, and indep) Pr info (event ) Pr(event info) Simple Forward Theory Subtle Inverse Inference ST Week 6 4

5 Bayes Rule Inverting the Conditioning Pr( Aand B) Pr( A B) Pr( B) Pr( B and A) Pr( B A) Pr( A) Pr( B) Pr( B A) Pr( A B) Pr( A) Multiple Possibilities B B OR B OR... B 1 2 Pr( B A) i Pr( A Bi)Pr( Bi) Pr( A B )Pr( B )... Pr( A B )Pr( B ) 1 1 n n n ST Week 6 5

6 Inversion 2 dice rolled; one is 6. What prob other is 6? 2 dice rolled; first is 6. What prob other is 6? ST Week 6 6

7 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" ( ). ST Week 6 7

8 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 students at 20 schools; hyper-texting 19.2% Such teens 43% more likely to have tried alcohol. ST Week 6 8

9 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? Evidence in favour of (stay with) chosen door ST Week 6 9

10 Serious Life Expectancy in Ireland Average age at death = 75 Average age at death = 80 Average age at death, given survival to 60, = 79 ST Week 6 10

11 Cond Prob for Lifetimes Knowledge of current age impacts uncertainty on age at death Probability Distribution Poss LiveTimes Corresp Probs Pr(death at end day 3, given alive at start day 3) ST Week 6 11

12 Cond Prob for Lifetimes Knowledge of current age impacts uncertainty on age at death Poss Lives Probs Expt: Choose random component Pr(death at end day 3) = 0.30 Pr(alive at start day 3) = 0.70 What event ident ity? Pr(death at end day 3, given alive at start day 3) Pr(death 'at' 3 AND alive 'at' 3) Pr(death 'at' 3) 0.3 = = 0.43 Pr(alive 'at' 3) Pr(alive 'at' 3) 0.7 Pr(death end day k, given alive day 3) ,,, for k 3, 4,5,6 resp ST Week 6 12

13 Odds and Weight of Evidence Odds (in favour of A) Odds Pr A Twice as likely Home Advantage Risk Factor Pr Pr 1 A A ST Week 6 13

14 Odds and Weight of Evidence Odds Rule Form for Evidence Pr( A) Pr( A) Pr( A E) Pr( E A) ; Pr( A E) Pr( E A) Pr( E) Pr( E) Pr( A E) Pr( E A) Pr( A) Pr( A E) Pr( E A) Pr( A) ST Week 6 14

15 Weight of Evidence Pr( A E) Pr( E A) Pr( A) Pr( A E) Pr( E A) Pr( A) A Car behind door chosen (eg 1); E Host opens door (eg 2) Prior Odds(A) = 1 2 Pr( E A) Pr( E A) A Posterior Odds(A) Posterior Pr(A) = ST Week 6 15

16 Weight of Evidence Pr( A E) Pr( E A) Pr( A) Pr( A E) Pr( E A) Pr( A) A Car behind door chosen (eg 1); E Host opens door (eg 2) Prior Odds(A) = 1 2 Pr( E A) Pr( E A) A Posterior Odds(A) Posterior Pr(A) = ST Week 6 16

17 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) ST Week 6 17

18 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) Events are T /F Define Define H X murderer E Blood group A left at crime scene Pr( E H )Pr( H ) Pr( H E) Pr( E H )Pr( H ) Pr( E) Pr( H)? Pr( H)? Pr( E H )? Pr( E H )? Pr( H E)? ST Week 6 18

19 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) Define Define H X murderer E Blood group A left at crime scene Pr( E H )Pr( H ) Pr( H E) Pr( E H )Pr( H ) Pr( E) Pr( H) Pr( H) 0.5 Pr( E H ) 1; Pr( E H ) Pr( H E) ST Week 6 19

20 Weight of Evidence Evidence Fusion Pr( A E1AND... AND En) Pr( E1 A) Pr( En A) Pr( A)... Pr( A E AND... AND E ) Pr( E A) Pr( E A) Pr( A) 1 n 1 if evidence indep given AA, n ST Week 6 20

21 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" ( ). Pr(2 Cot Deaths Normal Family) = Pr(Normal Family 2 Cot Deaths) Pr(Normal 2 Deaths) Pr(Normal ) Pr(2 Deaths Normal) = Pr(Not normal 2 Deaths) Pr(Not normal ) Pr(2 Deaths Not normal) ST Week 6 21

22 Serious Sally Clarke - Evidence Fusion Pr(Normal 2 Deaths) Pr(Normal ) Pr(1 Death Normal) = Pr(Not normal 2 Deaths) Pr(Not normal ) Pr(1 Death Not normal) nd Pr(2 Death Normal) nd Pr(2 Death Not normal) Formally Pr(2 Deaths Normal) = nd st Pr(2 Death 1 Death, Normal) Pr(1 Death Normal) ST Week 6 22

23 Background Overlaps completely with ST2351 ST Week 6 23

24 Decomposition via Conditional Probs Contestant chooses door 1, for example Car in fact behind each door with equal prob As car behind door 1 quiz master opens 2 or 3 with equal prob Contestant switches with prob 1 to Door: Goat Goat Car Contestant does not switch; remains with door 1 Car Car Goat Car Goat If car behind door 2 quiz master MUST open 3 ie prob = 1; sim ly if behind door 3 Prob wins = 2 3 ST Week

25 Inversion Survey of travellers in US finds: Pr A E Pr E A 20% have been to Europe; 15% have Amex card and have been to Europe; 55% have neither; What %age of Amex card holders have been to Europe? Amex Europe Europe Y N Y N Y 15% Y 15 Amex N 15% N 55 15% ST Week 6 25

26 Inversion Inverse Theory Prob ( 1 st Q, given 2 nd Q) Pr Q Q 1 2 PrQ Pr Q AND Q Pr Q Q Pr Q Pr Q Q Q Q Q Q Q Q Recall Pr Pr Pr Pr Pr Queens ST Week 6 26

27 Inversion Amex Europe Europe Y N Y N Y 15% Y 15 Amex N 15% N 55 15% Pr E A E Pr A AND E Pr A E Pr Pr A Pr Pr Pr Pr A E E A E E ST Week 6 27

28 Event Decomposition ( B OR B) Certain Event Pr( A) AND ( OR ) AND B OR A AND B A A B B A More generally B OR B OR... B Certain Event n A AND B OR A AND B... OR A AND B Pr( A) n A A AND A AND B OR B OR B 1 2 n ST Week 6 28

29 Event Identities Probability Monte Hall OR Choose Correct Door Choose Incorrect Door Certain Event Win by Staying Choose Correct Door Win by Switching Choose Incorrect Door Pr Win by Staying Pr( AOR B) Pr( A) Pr( B) Pr( A AND B) Important special case Pr( AOR B) Pr( A) Pr( B) when disjoint Addition Rule ST Week 6 29

30 Probability Rules Conditional Prob and Independence All computed probs: Sometimes Pr( A AND B) Pr( A B) Pr( B) Important special case Pr( A ANDB) Pr( A) Pr( B) when independent Pr Die 2AND Die Pr Die 2AND S need real world knowledge assumptions about real world useful to be explicit what - if Multiplication Rule ST Week 6 30

31 Chain Rule: Forward Theory Pr( Aand B) Pr( A B) Pr( B) Pr( A B) Extension Chain Rule Pr A AND B ANDC Pr( A AND B) C)Pr( C) But Pr( A AND B C) Pr( A B ANDC) Pr( B C) Pr( Aand B) Pr( B) Thus Pr A AND B ANDC Pr( A B ANDC)Pr( B C)Pr( C) Pr( A, BC, ) Pr( A B, C)Pr( B C)P r ( C) Special case Pr( A)Pr( B)Pr( C) if events probabilistically indep ST Week 6 31

32 Cards Forward Theory: Decompose By count By cond prob Prob 1 st two cards Queen? Define Pr Q, Q 1 2 Q ANDQ Pr Q Q Pr Q Simpler than combinatorics? Inverse Theory Prob ( 1 st Q, given 2 nd Q) Queens ST Week 6 32

33 Forward Theory: Decompose Cards Prob 2 nd card is Queen? Event Identity Q Q AND 2 2 whence Pr Q 2 ST Week 6 33

34 Forward Theory: Decompose Regular pack of cards; no replacement Focus Q 2 nd = 2 card is Q Decompose by considering Q Event Identity 1 Chance Tree Compute Pr( Q ) 2 Simulation equiv? Queens ST Week 6 34

35 Forward Theory: Decompose Password Draw 8 chars, unif at random, ABC XYZ Reject entire password if any dups within 8 Alt, reject char if same as any prev accepted char What is prob reject password (long run prop) ST Week 6 35

36 Password 8 from ABC XYZ (unif@rand, w/repl) Reject entire password if any dups within 8 N 7 N 6 N Pr Accept 8 N N N Pr Reject Forward Theory: Decompose Pr(at least one common birthday) = ST Week 6 36

37 Forward Theory: Decompose Password 8 from ABC XYZ (unif@rand, w/repl) Reject entire password if any dups within 8 Prob reject password Easier: Prob accept Define Seek A Accept first k chars from alphabet of N k Pr NOT A 8 A A AND A AND A A A AND A AND A Pr A A... AND A Pr A A... AND A Pr A1 Pr A A Pr A A... Pr A Event Identity Password OK =... Pr Pr N 7 N 6 N N N N Birthday problem ST Week 6 37

38 Forward Theory: Decompose Password 8 from ABC XYZ (unif@rand, w/repl) Reject entire password if any dups within 8 Prob reject password Easier: Prob accept Define A A AND A AND A Pr A A... AND A Pr A A... AND A... Pr A 8 Accept first kchars; Event Identity Password OK = A A AND A A AND A... AND A Pr Pr... Pr NOT A A k N 7 N 6 N N N N Birthday problem ST Week 6 38

39 Forward Recursion by Decomp: Dice Poss values of S are s 1,2, 6k k k k Pr S s p ; p 0, s 6( k 1) k s s S s S s ANDY any val k k k 1 Y y NDS s OR A y y1...6 k 1 k 1 Poss values of y 1,2, 6; Pr 1 Y y 6 k Y k p k s in terms of p k s p k1 ST Week 6 39

40 Conditional Independence A B C A C B C A B A AND B Formally Pr, Pr Pr Notation, A AND B A B Generalisation of Pr Pr Pr Example Markov Chain Y past Y Y Y Y Y t1 t1 t t1 t t1 Y Y Y Y Y Y Y Pr Pr Pr, Pr, Pr Pr t1 t1 t t1 t t1 t Subsys A using RAND() A Subsys C using RAND() C Subsys B using RAND() B Randomness in AC,BC dependent; eg AC large BC large (probably) Randomness in AC, BC cond indep, if we know the value of C ST Week 6 40