combining belief & desire basis of utility theory utility functions. multi-attribute utility...
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Combining Belief & Desire Basis of Utility Theory Utility Functions. Multi-attribute Utility Functions. Decision Networks Value of Information Expert Systems
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Rational Decision ◦ Based on Belief & Desire◦ Where uncertainty & conflicting goals exist
Utility◦ assigned single number by some Fn.◦ express the desirability of a state◦ combined with outcome prob. expected utility for each action
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Notation◦ U(S) : utility of state S◦ S : snapshot of the world◦ A : action of the agent◦ : outcome state by doing A◦ E : available evidence◦ Do(A) : executing A in current S
)(AResulti
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Expected Utility
Maximum Expected Utility(MEU)◦ Choose an action which maximizes agent’s expected
utility
i
ii AResultUADoEAResultPEAEU ))(())(,|)(()|(
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Rational preference◦ Preference of rational agent obey constraints◦ Behavior is describable as maximization of expected
utility
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Notation◦ Lottery(L): a complex decision making scenario
Different outcomes are determined by chance.◦ ◦ : A is preferred to B◦ : indifference btw. A & B◦ : B is not preferred to A
],1;,[ BpApL BA BA ~BA
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Constraints◦ Orderability
◦ Transitivity
◦ Continuity
)~()()( BAABBA
)()()( CACBBA
[ , ;1 , ] ~A B C p p A p C B
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Constraints (cont.)◦ Substitutability
◦ Monotonicity
◦ Decomposability
],1;,[~],1;,[~ CpBpCpApBA
],1;,[],1;,[( BqAqBpApqpBA
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Utility principle
Maximum Expected Utility principle
Utility ◦ Represents that the agent’s actions are trying to
achieve.◦ Can be constructed by observing agent’s
preferences.
BABUAU )()(
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iinn SUpSpSpU )(]),;;,([ 11
nF
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Utility◦ mapping state to real numbers◦ approach
Compare A to standard lottery u : best possible prize with prob. p u : worst possible catastrophe with prob. 1-p
Adjust p until
eg) $30 ~ L
pL
pLA ~
continue
death
0.9
0.1
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Utility scales◦ positive linear transform ◦ Normalized utility u = 1.0, u = 0.0 ◦ Micromort one-millionth chance of death
Russian roulette, insurance ◦ QALY quality-adjusted life years
0 )()(' 121 kwherekxUkxU
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Utility of Money◦ TV game
1million or 3 millions by chance 0.5
Example (Grayson, 1960)
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),(21
)(21
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000,000,1
000,000,3
k
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SUDeclineEU
SUSUAcceptEU
$800,000n to 000,1500$ from
)000,150log(09.2231.263)(
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nSU nk
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Money : does NOT behave as a utility fn.◦ Given a lottery L◦ risk-averse
◦ risk-seeking
U
$0
)()( )(LEMVL SUSU
)()( )(LEMVL SUSU
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Multi-Attribute Utility Theory (MAUT)◦ Outcomes are characterized by 2 or more attributes.◦ eg) Site a new airport
disruption by construction, cost of land, noise,….
Approach◦ Identify regularities in the preference behavior
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Notation◦ Attributes
◦ Attribute value vector
◦ Utility Fn.
,...,, 321 XXX
,...., 21 xxX
)](),...,([),...,( 111 nnn xfxffxxU
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Dominance◦ Certain (strict dominance, Fig.1)
eg) airport site S1 cost less, less noise, safer than S2 : strict dominance of S1 over S2
◦ Uncertain(Fig. 2)
Fig. 1 Fig.2
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Dominance(cont.)◦ stochastic dominance
In real world problem eg) S1 : avg $3.7billion, standard deviation : $0.4billion S2 : avg $4.0billion, standard deviation : $0.35billion
S1 stochastically dominates S2
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Dominance(cont.)
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-2
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Preferences without Uncertainty◦ preferences btw. concrete outcome values.◦ Preference structure
X1 & X2 preferentially independent of X3 iff Preference btw. Does not depend on
eg) Airport site : <Noise,Cost,Safety> <20,000 suffer, $4.6billion, 0.06deaths/mpm> vs. <70,000 suffer, $4.2billion, 0.06deaths/mpm>
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Preferences without Uncertainty(cont.)◦ Mutual preferential independence(MPI)
every pair of attributes is P.I of its complements. eg) Airport site : <Noise, Cost, Safety>
Noise & Cost P.I Safety Noise & Safety P.I Cost Cost & Safety P.I Noise: <Noise,Cost,Safety> exhibits MPI
Agent’s preference behavior ]))(()(max[
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Preferences with Uncertainty◦ Preferences btw. Lotteries’ utility◦ Utility Independence(U.I)
X is utility-independent of Y iff preferences over lotteries’ attribute set X do not depend on particular values of a set of attribute Y.
◦ Mutual U.I. (MUI) Each subset of attributes is U.I of the remaining
attributes. agent’s behavior( for 3 attributes) :multiplicative
Utility Fn. 131332322121332211 UUkkUUkkUUkkUkUkUkU
1313 UUkk
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Simple formalism for expressing & solving
decision problem
Belief networks + decision & utility nodes
Nodes◦ Chance nodes
◦ Decision nodes
◦ Utility nodes
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Airport Site
Air Traffic
litigation
Construction
UNoise
Death
Cost
CPT
utility table
Decision node
Utility node
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Airport Site
Air Traffic
litigation
Construction
U
Action-utility table
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Function DN-eval (percept) returns action
static D, a decision network
set evidence variables for the current state
for each possible value of decision node
set decision node to that value
calculate Posterior Prob. For parent nodes of the utility node
calculate resulting utility for the action
select the action with the highest utility
return action
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Idea◦ Compute value of acquiring each of evidence
Example : Buying oil drilling rights◦ Three blocks A,B and C, exactly one has oil, work k◦ Prior probabilities 1/3 each, mutually exclusive◦ Current price of each block is k/3◦ Consultant offers accurate survey of A. Fair price?
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Solution: Compute expected value of Information= Expected value of best action given information
- Expected value of best action without information
Survey may say “oil in A” with pdf 1/3or ‘no oil in A”( 2/3)
EV = 3
032
1
32
1
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2
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Notation◦ Current evidence E, Current best action ◦ Possible action outcomes Resulti(A)=Si ◦ Potential new evidence Ej
Value of perfect information (VPI)
i ii
A
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Nonnegative
Nonadditive
Order-Independent)()()()(),( ,, jEEkEkEEjEkjE EVPIEVPIEVPIEVPIEEVPI
kj
)()(),( kEjEkjE EVPIEVPIEEVPIj
0)( , jE EVPIEj
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a) Choice is obvious, information worth little
b) Choice is nonobvious, information worth a lot
c) Choice is nonobvious, information worth little
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Decision analysis◦ Decision maker
States Preferences between outcomes
◦ Decision analyst Enumerate possible actions & outcomes Elicit preferences from decision maker to determine the
best course of action
Advantage◦ Reflect preferences of the user, not system
◦ Avoid confusing likelihood and importance
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Determine the scope of the problem Lay out the topology Assign probabilities Assign utilities Enter available evidence Evaluate the diagram Obtain new evidence Perform sensitivity Analysis