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UNCERTAINTY
AMBIGUITY
Roger Cooke
Resources for the Future
Dept. Math, Delft Univ. of
Technology
April 15,16 2008
INDECISION
schedule
DAY 1
•
Start up
1.
Ambiguity, Uncertainty, Indecision
•
Break
2.
Expert Judgment for Quantifying Uncertainty
•
Expert Judgment Exercise
•
Lunch
3.
Bumper stickers for Expert Judgment
•
Break
4.
How to do an Expert Judgment Study
•
Round Table
DAY 2
•
Hands on EXCALIBUR
•
Break
•
Stakeholder elicitation exercise
•
UNIBALANCE demo
•
Lunch
5.
Utilities in Stakeholders Population
•
BREAK
•
Hands on UNIBALANCE
•
Round Table and evaluation
9:00
9:30
10:30
11:00
11:30
12:00
1:00
2:00
2:30
4:00
9:00
10:30
11:00
11:30
12:00
1:00
2:30
3:00
4:00
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Materials
• On CD:
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Materials
In Booklet
EJshortcourse1.ppt
EJshortcourse2.ppt
EJshortcourse3.ppt
EJshortcourse4.ppt
EJshortcourse5.ppt
EJcoursenotes-Classical-Model-Boilerplate
EJcoursenotes-Probability-Intro
EJcoursenotes-Theory-Rational-Decision
EJcoursenotes-Subj-Prob&RelFreq
EJcoursenotes-Proper-Scoring-Rules
EJcoursenotes-Review-Mathematics-Literature
EJcoursenotes-PI-definitions&theorems
EJcoursenotes-references
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Websites & Links
•
Radiation Protection Dosimetry 90: (2000)
http://rpd.oxfordjournals.org/cgi/content/short/90/3/295
•
NUREG EU Probabilistic accident consequence
uncertainty analysis
http://www.osti.gov/bridge/basicsearch.jsp
http://www.osti.gov/energycitations/basicsearch.jsp
•
EU Probabilistic accident consequence uncertainty
assessment using COSYMA
http://cordis.europa.eu/fp5-euratom/src/lib_docs.htm
•
RFF workshop expert judgment
http://www.rff.org/rff/Events/Expert-Judgment-Workshop.cfm
•
TU Delft Website
http://dutiosc.twi.tudelft.nl/~risk/
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How harmful is
100Gy gamma radiation
In 1 hr?
AMBIGUITY
Is John (5’11”) tall?
INDECISION
Is LOR better than EOR?
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What Is?
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What Means?
INDECISION
Whats best?
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Do measurements,
Quantify uncertainty
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Define concepts,
Domain of application
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Quantify utilities,
preferences
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Experts’ job
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Analyst’ job
INDECISION
Stakeholder/problem owners’
job
can’t remove uncertainty?
Uncertainty Analysis
NUREGCR-6545-Earlyhealth-VOL1.pdf
Using Uncertainty to Manage
Vulcano risk response
Aspinall et al Geol Soc _.pdf
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What is Uncertainty?
Probability?
Fuzzy sets?
Degree of possibility?
Certainty factors?
Dempster-Shafer Belief Functions?
Mathematical representation:
Axioms + Interpretation
Interpretation: aka
operational definitions
epistemic rules
rules of correspondence
etc etc
squizzel.pdf
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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 language not containing
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“degree of possibility” is BOB committed?
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Objective and Subjective Probability
EJCourseNotes-Probability-Intro.doc
•
Probability formalism is Kolmogorov’s
axioms, for all events A,B:
1. 0  P(A)  1
2. P(A’) = 1 – P(A)
3. If A  B =   P(AB) = P(A)+P(B)
•
These can be interpreted either
– OBJECTIVELY: Limit Relative Frequency,
OR
– SUBJECTIVELY: Partial Belief
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Objective: Limit relative
frequency
• Naïve
– Let A1, A2…be “independent trials of A”, then P(A) =
lim (#occurrences in N trials / N)
• Need probability to define “independent trials”
• Von Mises (1919)
– P(outcome i) = lim relative freq of i in a “kollectif” of
outcomes, i.e. random sequence
• Need definition of “random sequence”
• Kolmogorov, Martin-Lof, Schnorr, etc. (60’s-70’s)
– Random sequence is one which passes all recursive
statistical tests  is not predictable by any “decidable
rule”.
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Examples
Heads with bent coin:
N N N N Y Y N N N Y N N Y N N N Y Y N N
PROBABILITY {Heads} = 3/10: THIS IS RANDOM SEQUENCE
Thruster Failure on previous tests
Y Y N Y Y N Y N Y N N N N N N N N N N N
PROBABILITY OF FAILURE  3/10: NOT RANDOM SEQUENCE
USA wins of previous World Cup Soccer championships
N N N N N N N N N N N N N N N N N N N N
PROBABILITY OF USA WIN
 0:
Not a RANDOM SEQUENCE
Subjective: Degree of Partial
Belief
(Ramsey 1926, Borel, DeFinetti 1937, von Neumann & Morgenstern
1944, Savage, 1954)
• Measure partial belief
• See if it satisfies axioms of probability
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Operational definition: Subjective
probability
Consider two events:
F: France wins next World Cup Soccer tournament
US: USA wins next World Cup Soccer tournament.
Two lottery tickets:
L(F): worth $10,000 if F, worth $100 otherwise
L(US): worth $10,000 if US, worth $100 otherwise.
John may choose ONE .
John's degree belief (F)  John’s degree belief (US)
is operationalized as
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John chooses L(F) in the above choice situation
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If your preferences satisfy ‘principals of
rationality’:
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B: Belgium wins next World Cup Soccer tournament.
L(F) > L(US); L(US) > L(B);  L(F) > L(B) ??
L(F) > L(US)  L(F or B) > L(US or B) ??
(plus some technical axioms)
Then (Fundamental Theorem of Decision Theory)
There is a UNIQUE probability P which represents degree of belief:
DegBel(F) > DegBel(US)  P(F) > P(US)
AND a Utility function, unique op to 0 and 1, that represents values:
L(F) > L(US)  Exp’d Utility (L(F)) > Exp’d Utility (L(US))
PROOF (4 hrs) EJCoursenotes-Theory-Rational-Decision.doc
INDECISION
Can subjective probabilities be
relative frequencies???
A1, A2…….An…..: yes-no experiments
S’pose partial belief independent of order:
P{A1=Y, A2= N, A3=N} =
P{A1=N, A2= N, A3=Y}
THEN (barring pathological case)
P(An+1= Y | A1=Y,A2= N, A3=N…An=N) n  #Y / n
PROOF: combinatorics (20 min),
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EJCoursenotes-SubjectiveProb&RelFreq.doc
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To clarify
• Subjective probabilities can = relative frequencies
• 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
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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CAN fuzziness represent
uncertainty?
μman(Quincy) = μwoman(Quincy) = ½

μman AND woman(Quincy) =
Min {μman(Quincy), μwoman(Quincy)} = ½
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Lets have a break
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