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Transcript
Subjective probability
• Often, we estimate likelihood of outcomes
of uncertain events using judgment
• Examples:
– Likelihood of major earthquake (7.5-8 on
Richter scale) in Southern California over next
30 years?
– Flipped a coin that has landed on floor. Have
not seen coin. What is the likelihood that it is
heads?
1
Interpretations of probability
Probability
Long-term relative
frequency
Subjective:
Degree of decision-maker’s
belief that outcome of uncertain
will of occur
2
Premise
• We can always assess a decision-maker
uncertainty using probabilities.
• To assess a decision-makers probability
observe his/her attitude toward accepting
bets about the outcome in question.
3
Assessing subjective probability
• Assume risk neutral decision maker;
his/her utility of an amount of money is
proportional to the amount
4
Lottery ticket
A occurs
Receive $1
A does not occur
Receive $0
5
Assessing probability of A
• If a decision-makers subjective probability
of A is p, then he/she is willing to pay p of
the ticket AND
• He/she is willing to sell a ticket for p.
6
Practical ways to assess subjective
probabilities
1. Ask directly decision-maker
2. Ask decision-maker what bets she/he is
willing to place
3. Compare lottery involving the outcome
whose probability you want to estimate
with reference lottery whose probability
mechanism is known
7
Asking decision-maker what bets
she/he is willing to place
A occurs
X
Bet for A
A does not occur
A occurs
Bet against A
-Y
-X
A does not occur
Y
Find amounts X and Y that make the two bets equivalent.
Y
Then
P( A ) 
X Y
8
Asking decision-maker what bets
she/he is willing to place
Procedure
• Let X+Y=$1,000. Start with X=$999.99 Then
the decision-maker would prefer betting for A
than against it.
• Reduce incrementally X each time asking
decision maker which bet he/she prefers. Stop
when the decision maker becomes indifferent for
the two bets.
• Find probability of A
• Example: X=$100 and Y=$900 makes decision
maker indifferent for two bets. Then P(A)=0.9.
9
Compare lottery involving the outcome whose
probability you want to estimate with reference
lottery whose probability mechanism is known
A occurs
$1000
Lottery
A does not occur
A occurs
Reference lottery
1-p
Free hamburger
$1000
(p)
A occurs
Free hamburger
(1-p)
p
Find probability p that makes reference lottery equivalent to first lottery.
10
Assessing continuous probabilities
Assess few values of cumulative probability distribution
function, for different values of the random variable.
Connect with smooth curve
Cumulative probability distribution of grade
1
P(grade80)
0
80
100
Grade, z
11
Decomposition for assessing
subjective probabilities
• Break task of assessing probability into
smaller tasks. Outcomes whose
probabilities are easier to assess.
• Example: probability of accident?
Human
error
Accident
No accident
Accident
No human
error
No accident
12
Decomposition for assessing
subjective probabilities
P(Accident) = P(Accident/Human
Error)P(Human Error)+ P(Accident/No
Human Error)[1-P(Human Error)]
It could be easier to assess probabilities on
right hand side of equation than probability
of an accident
13
Coherence
• Probabilities (subjective or long term
relative frequencies) must cohere
– Must be between 0 and 1
– P(AB)=P(A)+P(B) if A and B are disjoint
– P(A)+P(AC)=1
AC
A
14
Why coherence
• Because if a decision makers subjective
probabilities do not cohere he/she may
incur sure loss; a competitor can set us a
Dutch book to drain up his/her account
• Dutch book: heads, competitor wins; tails,
decision maker losses.
15