Download Subjective probability

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?
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Interpretations of probability
Probability
Long-term relative
frequency
Subjective:
Degree of decision-maker’s
belief that outcome of uncertain
will of occur
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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.
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Assessing subjective probability
• Assume risk neutral decision maker;
his/her utility of an amount of money is
proportional to the amount
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Lottery ticket
A occurs
Receive $1
A does not occur
Receive $0
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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.
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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
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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
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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.
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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.
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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
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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
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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
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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
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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.
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