Download Induction and Decision Trees

Axioms
•Let W be statements known to be true in a domain
•An axiom is a rule presumed to be true
•An axiomatic set is a collection of axioms
•Given an axiomatic set A, the domain theory of A, domTH(A)
is the collection of all things that can be derived from A
Axioms (II)
•A problem frequently studied by mathematicians: Given W
can we construct a (finite) axiomatic set, A, such that
domTH(A) = W?
•Potential difficulties:
Inconsistency:
W  domTh(A)
Incompleteness:
domTh(A)  W
•Theorem (Goedel): Any axiomatic set for the Arithmetic is
either inconsistent and/or incomplete
Recap from Previous Class
•First-order logic is not sufficient for many problems
•We have only a degree of belief (a probability)
Decision Theory = probability theory + utility theory
•Probability distribution
Today!
•Expected value
•Conditional probability
•Axioms of probability
•Bruno de Finetti’s Theorem
Utility of A Decision
CSE 395/495
Resources:
–Russell and Norwick’s book
Two Famous Quotes
“To judge what one must due to obtain a good or avoid an
evil, it is necessary to consider not only the good and the evil
itself, but also the probability that it happens or not happen”
Arnauld, 1692
“… so they go in a strange paradox, decided only to be
undecided, resolved to be irresolute, adamant for drift, solid
for fluidity all powerful to be impotent”
Churchill, 1937
Utility Function
•The utility captures an agent’s preference
U: States  [0,)
•Given an action A, let Result1(A), Result2(A), … denote the
possible outcomes of A
•Let Do(A) indicates that action A is executed and E be the
available evidence
•Then, the expected utility EU(A|E):
EU(A|E) = i P(Resulti(A) | E, Do(A)) U(Resulti(A))
Principle of Maximum Expected Utility
(MEU)
An agent should choose an action that maximizes EU
Suppose that taken actions update
probabilities of states/actions.
Which actions should be taken?
?
?
?
1. Calculate probabilities of current state
2. Calculate probabilities of the actions
3. Select action with the highest
expected utility
MEU says choose A for state S such that for any other action A’ if E
is the known evidence in S, then EU(A|E)  EU(A’|E)
MEU Doesn’t Solve All AI Problems
EU(A|E) = i P(Resulti(A) | E, Do(A))U(Resulti(A))
Difficulties:
•State and U(Resulti(A)) might not be known completely
•Computing P(Resulti(A) | E, Do(A)) requires a causal
model. Computing it is NP-complete
However:
It is adequate if the utility function reflects the performance
by which one’s behavior are judged
Example? Grade vs. knowledge
Lotteries
•We will define the semantics of preferences to define the
utility
•Preferences are defined on scenarios, called lotteries
•A lottery L with two possible outcomes: A with
probability p and B with probability (1– p), written
L =[p, A; (1– p), B]
•The outcome of a lottery can be an state or another lottery
Preferences
Let A and B be states and/or lotteries, then:
•A  B
denotes A is preferred to B
•A ~ B
denotes A is indifferent to B
•A  B
denotes either A  B or A ~ B
Axioms of the Utility Theory
• Orderability
•Transitivity
•Continuity
A  B or B  A or A ~ B
If A  B and B  C then A  C
If A  B  C then p exists such that
[p, A; (1– p), C] ~ B
•Substitutability
If A ~ B then for any C
[p, A; (1– p), C] ~ [p, B; (1– p), C]
Axioms of the Utility Theory (II)
•Monotonicity
If A  B then p  q iff
[p, A; (1– p), B]  [q, A; (1– q), B]
•Decomposibility (“No fun in gambling”)
[p, A; (1– p), [q, B; (1– q), C] ] ~
[p, A; (1– p)q, B ; (1– p)(1– q), C ]
Axioms of the Utility Theory (III)
•Utility principle U: States  [0,)
A  B iff
U(A) > U(B)
A ~ B iff
U(A) = U(B)
•Maximum Expected Utility principle
MEU([p1,S1; p2,S2; … ; pn,Sn]) = i piU(Si)
Example
Suppose that you are in a TV show and you have already
earned 1’000.000 so far. Now, the presentator propose you
a gamble: he will flip a coin if the coin comes up heads you
will earn 3’000.000. But if it comes up tails you will loose
the 1’000.000. What do you decide?
First shot:
U(winning $X) = X
MEU([0.5,0; 0.5,3’000.000]) = 1’500.000
This utility is called the expected monetary value
Example (II)
If we use the expected monetary value of the lottery does it
take the bet?
Yes!, because:
MEU([0.5,0; 0.5,3’000.000]) = 1’500.000 >
MEU([1,1’000.000; 0,3’000.000]) = 1’000.000
But is this really what you would do?
Not me!
Example (III)
Second shot:
Let S = “my current wealth”
S’ = “my current wealth” + $1’000.000
S’’ = “my current wealth” + $3’000.000
MEU(Accept) = 0.5U(S) + 0.5U(S’’) = 7.5
MEU(Decline) = U(S’) = 8
If U(S) = 5, U(S’) = 8, U(S’’) = 10, would you accept the bet?
No!
U
$
Human Judgment and Utility
•Decision theory is a normative theory: describe how agents
should act
•Experimental evidence suggest that people violate the axioms
of utility Tversky and Kahnerman (1982) and Allen (1953):
Experiment with people
Choice was given between A and B and then
between C and D:
A: 80% chance of $4000 C: 20% chance of $4000
B: 100% chance of $3000 D: 25% chance of $3000
Human Judgment and Utility (II)
•Majority choose B over A and C over D
If U($0) = 0
MEU([0.8,4000; 0.2,0]) = 0.8U($4000)
MEU([1,3000; 0,4000]) = U($3000)
Thus, 0.8U($4000) < U($3000)
MEU([0.2,4000; 0.8,0]) =
0.2U($4000)
MEU([0.25,3000; 0.65, 0]) = 0.25U($3000)
Thus, 0.2U($4000) > 0.25U($3000)
Thus, there cannot be no utility function consistent with
these values
Human Judgment and Utility (III)
•The point is that it is very hard to model an automatic agent
that behaves like a human (back to the Turing test)
•However, the utility theory does give some formal way of
model decisions and as such is used to support user’s
decisions
•Same can be said for similarity in CBR
Homework
You saw the discussion on the utility relative to the
“talk show” example. Do again an analysis of the
airport location that was given last class but this
time around your discussion should be centered
around how to define utility rather than the
expected monetary value.