ST2004 Week 7 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.
SpamCancer ScreeningLawExams
ST2004 Week 7 2
Probability RulesConditional Prob and Independence
Important special Pr( ) Pr( | ) Pr( )
Pr( ) Pr( ) Prcase
( )
Aand B A B B
Aand B A B when independent
Multiplication Rule
All computed probs: depend on real world knowledgeassumptions about real world
Sometimes Useful to be explicitWhat - 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 = 75
Average age at death = 80
Average age at death,given survival to 60, = 79
http://understandinguncertainty.org/node/272
ST2004 Week 7 7
Light
Metro Wed 10 Nov 2010Teens 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 10
Cond Prob for LifetimesKnowledge of current age impacts uncertainty on age at death
Probability DistributionPoss LiveTimes 1 2 3 4 5 6Corresp Probs 0.1 0.2 0.3 0.3 0.05 0.05
Pr(death at end day 3, given alive at start day 3)
ST2004 Week 7 12
Conditional Prob: Chain Rule
Extension Chain Rule
Pr Pr( ) | ) Pr( )
Pr(
Pr( )Pr( ) P
| ) Pr( | ) Pr( | )
Pr Pr( | ) Pr( | ) Pr(
r( | ) Pr( ) Pr( | )Pr( )
Pr( , , ) Pr( | , ) Pr( | ) P (
)
r )
A ANDB ANDC A ANDB C C
But A ANDB C A B AND
Aand BAand B A B B A BB
A
C B C
Thus A ANDB ANDC A B ANDC
B C
B
A B C B C C
Sp
C C
Pr( ) Pr( ) Pr( )ecial case if events probabilistically iA B C ndep
ST2004 Week 7 13
Conditional Prob: Chain Rule
th = Ace on i drawSeek Pr (3A in 3 Draws)
Event Identity
Prob EquationPr 3A in
Defi
3 d
n
ws
e
ra
iA
ST2004 Week 7 17
Decomposition via Conditional Probs
1 2
1 2
1 2
( )
Pr( )
................
................
...
Pr( )
n
n
n
A A AND B OR B A AND B OR A ANDB
A
B OR B OR B
A A AND AAND B OR B OR B
A ANDB OR A ANDB OR A ANDB
A
ST2004 Week 7 18
Decomposition via Conditional Probs
nd
Regular pack of cards; no replacement
Pr( 2 card is Q)
Chance Tree
ST2004 Week 7 21
Decomposition via Conditional Probs
Define Score on die; no heads in tossesS F S
Chance Tree
ST2004 Week 7 23
Brain TeasersMonty 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 24
Decomposition by Conditional Sim
Contestant always chooses Door 1Car behind random door – equal probsHost actions Don’t Switch Do Switch PrizeCar behind
123
Pseudo Code
ST2004 Week 7 25
Decomposition by Conditional SimContestant Always Chooses Door 1Car behind Random DoorGiven Car behind Door
1 2 3 Car behind randomly chosen door2
Pr Host opens Host Opens Door 31 0 0.5 0.52 0 0 1 Relevant Cum probs3 0 1 0 1 2 3
0 0 0 1Cum Probs
1 0 0 0.5 12 0 0 0 1 Win3 0 0 1 1 Don't switch 1 Goat
Switch to door 2 Car
Final Choice Reps
1 Goat Car2 Car Goat3 Car Goat4 Goat Car 645 3555 Goat Car6 Goat Car
Don't switch
Do Switch
Don't switch
Do Switch
Number of Times win Car
ST2004 Week 7 26
Decomposition via Conditional ProbsChance Tree
2
1
1 2
3
3
2
3
Contestantchooses door1, for example
3
2
2
3
Car
Car
Goat
Goat
Contestantdoes not switch;
Car
Car
Goat
Goat
Contestantswitches
Prob wins =
Carbehind
Hostopens
ST2004 Week 7 29
Bayes Rule
Pr( ) Pr( | ) Pr( ) Pr( ) Pr( | ) Pr( )Pr( )
Pr( | ) Pr( | )Pr( )
Aand B A B B Band A B A AB
B A A BA
Inverting the Conditioning
Multiple Possibilities
1 2
1 1
................Pr( | )Pr( )Pr( | )
Pr( | )Pr( ) ....... Pr( | )Pr( )
n
i ii
n n
B B OR B OR BA B BB A
A B B A B B
ST2004 Week 7 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 7 34
Brain Teaser
Roll regular die: note score NToss fair coin N times
Observe no heads.What now Pr(scored 2)?Pr(scored 2|no heads)?
ST2004 Week 7 35
Bayes Rule
Odds Rule Form for EvidencePr( ) Pr( )Pr( | ) Pr( | ) ; Pr( | ) Pr( | )Pr( ) Pr( )
Pr( | ) Pr( | ) Pr( )Pr( | ) Pr( | ) Pr( )
B AA E E A A E E AA E
A E E A AA E E A A
Evidence Fusion
1 1
1 1
Pr( | ..... ) Pr( | )Pr( | ) Pr( )....Pr( | ..... ) Pr( | ) Pr( | ) Pr( )
if evidence indep given ,
n n
n n
A E AND AND E E AE A AA E AND AND E E A E A A
A A
ST2004 Week 7 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).
218543Pr(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)
ST2004 Week 7 39
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 1Told family has two children; one is a daughterProb other is daughter?Told family has two children; ring bell – girl opensProb other also a girl?
ST2004 Week 7 43
Spam Exam QA 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
Pr( ),Pr( | ),Pr( | )S F S F S
Define Reported as spamPr( ) 0.01Pr( | ) Pr( | ) Pr(False Negative) = 0.02
Pr( | ) Pr( | ) Pr(False Positive) = 0.05
RSR S F S
R S F S
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