Refresher: Foundations of Decision Theory · Decision Theory L.J. Savage 1954: • Choice behavior...
Transcript of Refresher: Foundations of Decision Theory · Decision Theory L.J. Savage 1954: • Choice behavior...
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Refresher:Foundations of
Decision TheoryRoger Cooke
Chauncey Starr Senior FellowResosurces for the Future
March 14, 1006
•Decision theory•Uncertainty•Preference
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Uncertainty is that which disappears when we become
certain
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Decision Theory L.J. Savage 1954:
• Choice behavior of a ‘rational individual’can be represented as expected utility with– Unique probability measure– Unique Utility up to ‘0’and ‘1’.
• BUT groups are not rational individuals...aim for rational consensus
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Paradox of Majority Preference1/3 prefer Mozart > Hayden > Bach 1/3 prefer Hayden > Bach > Mozart1/3 prefer Bach > Mozart > Hayden
THEN 2/3’s prefer
Bach > MozartMozart > HaydenHayden > Bach
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Distinguish and Separate
• Uncertainty wrt ‘what is the case’
• Values, utilities, preferences
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Who does what?
Uncertainty ⇒ Domain experts
Utility, preferences ⇒ Policy makers, stakeholders, voters
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UncertaintyWhy represent uncertainty as
probability??
Foundations: Axioms + Interpretation
Interpretation: akaoperational definitionsepistemic rulesrules of correspondenceetc etc
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Operational Definitions• The philosophy of science: semantic analysis:
Mach, Hertz, Einstein, Bohr
• A Modern rendering:
IF BOB says
“The Loch Ness monster exists with degree of possibility 0.0731”
to which sentences in the natural languagenot containing “degree of possibility”is BOB committed?
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Operational definition: Subjective probability
Consider two events:F: France wins next World Cup Soccer tournamentU: USA wins next World Cup Soccer tournament.
Two lottery tickets:L(F): worth $10,000 if F, worth $100 otherwiseL(U): worth $10.000 if U, worth $100 otherwise.
John may choose ONE .
John's degree belief (F) >= John’s degree belief (U)
is operationalized as
John chooses L(F) in the above choice situation
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Ask yourself:B: Belgium wins next World Cup Soccer tournament.
L(F) > L(U); L(U) > L(B); ⇒ L(F) > L(B) ??L(F) > L(U) ⇒ L(F or B) > L(U or B) ??(plus some technical axioms)
Then there is a UNIQUE probability P which represents degree of belief (Savage’s representation theorem):
DegBel(F) > DegBel(U) ⇔ P(F) > P(U)
This can be measured observing INDIVIDUAL choice behavior
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Operational definition, frequentist probability
“Prob(A) = 0.873” means“The relative frequency of occurence of A in a
REFERENCE CLASS is 0.873.”
Reference class must be a “collective”, a sequence of ‘trials’ for A, such that every ‘decidable subsequence’ has same limit rel. frequency. (von Mises, Kolmogorov)
MUST SPECIFY REF. CLASS
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To clarify• You can be uncertain about a limit rel. frequency
• You can learn about a rel. freq. thereby reducing your uncertainty
• You can quantify your uncertainty conditional on, say, X, and be uncertain about X
• You cannot be uncertain about your uncertainty in any other useful sense.
“my uncertainty in success is 0.7, but my uncertainty in my uncertainty is 0.5, and my uncertainty in my uncertainty of my uncertainty is 0.3....”DON’T GO THERE
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Other interpretations of Probability axioms
• Classical interpretation (Laplace) ‘ratio of favorable cases to all equi-possible cases’
• Logical Interpretation (Keynes, Carnap) ‘partial logical entailment’
Neither were able to provide successful operational definitions.
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Alternative representations of uncertainty have no foundation
Fuzzy sets: many axiomatizations, no operational definitions
Degree of Possibility: no operational definitions(see however Eur. J. of Oper. Res. 128, 459-478.p 477).
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EG if µA(x) = fuzzy membership of x in A
DOES µwinworldcup(FRANCE) > µwinworldcup(USA)
ENTAIL
L(F) > L(U) ????
Does it entail anything not involving “fuzzy”?
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CAN fuzziness represent uncertainty?
µman(Quincy) = µwoman(Quincy) = ½⇒µman AND woman(Quincy) =Min {µman(Quincy), µwoman(Quincy)} = ½
L(Quincy is a man) ~ L(Quincy is a hermaphrodite) ??
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EJ for RATIONAL CONSENSUS:
Parties pre-commit to a method which satisfies necessary conditions for scientific method:
Traceability/accountabilityNeutrality (don’t encourage untruthfulness)Fairness (ab initio, all experts equal)Empirical control (performance meas’t)
Withdrawal post hoc incurs burden of proof.
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What is a GOOD subjective probability assessor?
• Calibration, statistical likelihood– Are the expert’s probability statements
statistically accurate?• Informativeness
– Probability mass concentrated in a small region, relative to background measure
• Nominal values near truth• ?
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Expert elicitation techniques
• Delphi• Nominal group techniques• Group nomination• Team building, decision conferencing, etc
Key question: How do we measure performance
Credibility via performance, period.
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Preference:Stakeholder Preference Theory
Simple paired comparisons (consumer research)
MCDMMAUTAHPEtc etc
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Consumer sampling ⇒valuation of competing products
Ask random consumer:Do you prefer
APPLES to BANANAS?APPLES TO ORANGESBANANAS TO ORANGES?ETC
Find a value function V(x) that represents preferences
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Paired Comparison Data Analysis
• Is each customer’s preference non-random?
• Is the agreement between customers non-random?– Coefficient of agreement– Coefficient of concordance
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Thurstone’s model
-6 -4 -2 0 2 4 6
Apples
Bananas
Oranges
Assume values are normally distributed over stakeholders:
Find relative placement that reproduces preferences
V(Orange) V(Banana) V(Apples)
Values are determined up to “0 and 1”.
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Bradley-Terry
Assume that pairwise preference is related to value V(x) as:
V(APPLES)%(APPLES > BANANAS) =
V(APPLES)+V(BANANAS)
= PROB ( apples | apples or bananas )
Values determined up to a constant
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Hands-on
• Expert elicitation – Air quality or– SARS
• Stakeholder preference: 6 fuel policies