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Transcript
```CHAPTER 8: SUBJECTIVE PROBABILITY
BASIC TENET OF DECISION ANALYSIS
 judgments about uncertainty can be expressed in probabilities
UNCERTAINTY AND PERSONAL DECISIONS
 next job
 future job opportunities
 location to live
 medical decisions
 investment decisions
UNCERTAINTY AND PUBLIC POLICY
 Hurricane warning
 Earthquake prediction
 Environmental risk
 Greenhouse effect
 Diseases, e.g., AIDS
 Defense spending
1
PROBABILITY: A FREQUENCY VIEW
 probability based on long-run frequency
- results subjectively extrapolated to current situation
 results may be one-of-a-kind
- probability of nuclear power accident for a new nuclear power plant design
PROBABILITY: A SUBJECTIVE INTERPRETATION
 subjective view: probability represents an individual's belief that an event will
occur
- ALWAYS based on a state of information
- update when the state of information changes
 probability assessment is art with some scientific basis
ASSESSING DISCRETE PROBABILITIES: 2 APPROACHES (3rd in Text)
1. Direct Question
- what is the probability you will get an A in MAT 643?
- requires DM who is comfortable providing probabilities
- can use this method after using the 2nd method a couple times
2
2. Have decision-maker compare with a reference lottery
+ \$1000
A
Bet 1
Not A
Bet 2
- \$ 1000
+ \$1000
p
1-p
- \$1000
 use probability wheel to find p such that you are indifferent between Bet 1 and
2
- advantage: probabilities won't be even
Last Step
- check for consistency
QUESTIONS:
1. Did our definition of an “A” satisfy the clarity test?
2. How can we check for consistency?
ASSESSING CONTINUOUS PROBABILITIES
 two strategies for assessing continuous CDF
3
 consider the age of the a volunteer student
1. Set outcomes and assess probabilities
 specify age, x, and assess p = P[age < x]
Age < =x
Bet 1
Age > x
Bet 2
+ \$1000
- \$ 1000
+ \$1000
p
- \$1000
1-p
2. Set probabilities and determine outcomes
 specify P[age < x], and determine x
Bet 1
+ \$1000
Age < =x
- \$ 1000
Age > x
Bet 2
+ \$1000
0.25
- \$1000
0.75
 alternative approaches
4
- quartiles: 0.0, 0.25, 0.5, 0.75, 1.0
- 0.05 and 0.95 quartiles, 0.25 and 0.75, then 0.5
QUESTION: Which of the two approaches would be the best?
CONVERTING CONTINUOUS DISTRIBUTIONS TO DISCRETE
DISTRIBUTIONS
(DPL NOTE: DPL TAKES CARE OF THIS. BEST TO USE 3-4 OUTCOMES.)
 see the decision tree for the outcomes and probabilities
1. Extended Pearson-Tukey Method (Keefer & Bodily, 1983)
0.185
0.05 fractile
0.63
0.185
median
0.95 fractile
 best for symmetric distributions
 OK for asymmetric distributions
 requires less assessments
2. Moment Matching
 Used by DPL
5
 Example Normal(10,2) with 3 outcomes
Normal_Variable
[10]
Low
.167
Nominal
.667
High
.167
[6.54]
6.54
[10]
10
[13.5]
13.5
 Example: Normal (10,4) with 3 outcomes
Normal_Variable
[10]
Low
.167
Nominal
.667
High
.167
[3.07]
3.07
[10]
10
[16.9]
16.9
 QUESTION: What's the difference?
 Example: Normal (10,4) with 5 outcomes
6
Normal_Variable
[10]
Lowest
.011
[-1.43]
-1.43
Lower
.222
[4.58]
4.58
Nominal
.533
[10]
10
High
.222
[15.4]
15.4
Highest
.011
[21.4]
21.4
 QUESTION: What's the difference?
PITFALLS: HEURISTICS AND BIASES
 probability assessment is not easy
 heuristics: cognitive techniques we use to make probability assessments
- Tversky & Kahneman, 1974
TOM W. EXAMPLE, p. 281
Representativeness
 make assessments on similarity and ignore base rates
 use of stereotypes
 misunderstanding of random processes
7
 regression to the mean
- extreme cases will be followed by less extreme cases
Availability
 ease by which we can recall similar events from memory
 recent events may bias our assessments
 unbalanced reporting can bias assessments
 illusory correlation: events that happen together may be assumed to be related
 choose initial anchor and then adjust
 if we begin with median, tend to underadjust
Motivational Bias
 people tend to manage our expectations
Overcoming Heuristics and Biases
 individuals can become good at probability assessment
 awareness of heuristics and biases helps
8
 structured techniques help
 if assessment is difficult, try decomposition
DECOMPOSITION & PROBABILITY ASSESSMENT
 break probability assessment into smaller, more assessable probabilities
 use influence diagram
 examples
- probability of nuclear accident in a new reactor design
- probability of use of a nuclear weapon by a specific country that supports
terrorist activities
COHERENCE & THE DUTCH BOOK (MONEY PUMP)
 subjective probabilities must obey axioms & laws of probability
- coherent
 de Finetti's Dutch Book Theorem
- if a person's probabilities are not coherent, you can set up a Dutch Book
against them
- Dutch Book: a series of bets that, in the long run,
you are guaranteed
to win. The longer you play, the more you win. (Money Pump)
9
EXAMPLE: suppose you assessed P[H]= 0.4 and P[T]=0.5
QUESTION: How many times would you like to play this bet?
Bet 1
Not H
H
Bet 2
+ \$ 40
- \$ 60
+ \$ 50
T
Not T
- \$ 50
 incoherence can be exploited
 de Finetti showed that only possible not to be exploited if subjective
probabilities were coherent
SUMMARY
 subjective probability
 assessing probabilities
- discrete
- continuous
10
 converting continuous to discrete
 heuristics & biases
- representativeness
- availability