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EE562
ARTIFICIAL INTELLIGENCE
FOR ENGINEERS
Lecture 14, 5/23/2005
University of Washington,
Department of Electrical Engineering
Spring 2005
Instructor: Professor Jeff A. Bilmes
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Uncertainty
Chapter 13
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Outline
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Limitations of logic (Gödel)
Uncertainty
Probability
Syntax and Semantics
Inference
Independence and Bayes' Rule
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Reminder
• HW4 Due Today!!
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General Logics
• Ontology:
– science or study of being, which problems are being
investigated, the kinds of abstract entities that are to be admitted
to a language system, things that exist in a system.
• Epistemology:
– study of methods and grounds of knowledge, its limits, and its
validity, theories of knowledge.
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Example knowledge base
• The law says that it is a crime for an American to sell
weapons to hostile nations. The country Nono, an
enemy of America, has some missiles, and all of its
missiles were sold to it by Colonel West, who is
American.
•
• Prove that Col. West is a criminal
•
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Example knowledge base
(in logic form)
... it is a crime for an American to sell weapons to hostile nations:
American(x)  Weapon(y)  Sells(x,y,z)  Hostile(z)  Criminal(x)
Nono … has some missiles, i.e., x Owns(Nono,x)  Missile(x):
Owns(Nono,M1) and Missile(M1)
… all of its missiles were sold to it by Colonel West
Missile(x)  Owns(Nono,x)  Sells(West,x,Nono)
Missiles are weapons:
Missile(x)  Weapon(x)
An enemy of America counts as "hostile“:
Enemy(x,America)  Hostile(x)
West, who is American …
American(West)
The country Nono, an enemy of America …
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Enemy(Nono,America)
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Forward chaining proof
Note: can be seen as simply a list of logic sentences. So, any proof is
just a other facts that can be deduced from the KB to achieve new
knowledge.
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Gödels Incompleteness Theorem
(general idea)
• Each sentence in logic has a length (say number
of symbols).
• We can enumerate all the sentences in any logic
system with a finite number of different types of
function symbols (e.g., 8, 9, +, £,, x, etc.)
• First enumerate all sentences of length 1 (a finite
number of them), then number all of length 2
(again, a finite number), etc.
• Thus, there are a countable number of possible
sentences in logic.
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Gödels Incompleteness Theorem
(general idea)
• Let #() be the number of sentence 
• Let #-1(j) be the sentence for number j
• Note that all proofs have numbers also.
– i.e., a sequence of sentences also has a
number, and any proof can be seen as a
sequence of sentences.
• Let A be a set of true sentences.
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Gödels Incompleteness Theorem
(basic idea)
• Let A be a set of true premises (e.g., could be
thought of as a KB, i.e., A=KB)
• Consider the following sentence, represented by
the function (j,A),
– when we use only premises in A, then:
• In words: there is no sentence i that proves j,
when we start using only A.
• This sentence (j,A), like all others, is one that
can be either true or false under A.
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Gödels Incompleteness Theorem
(basic idea)
• Repeated: (j,A) is the sentence:
– when we use only premises in A, then:
• Now consider the sentence defined as:
•  is the sentence that states its own
unprovability when using only A. The
sentence , again, can be true or false.
• In A, assume  true, then it is false
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• In A, assume  false,
then it is true.
Gödels Incompleteness Theorem
(basic idea)
• Assume  follows from A, but then  is false
(since  says it doesn’t follow from A).
• But then if  is false, then  does follow from A,
which means A must be false (violating our
premise)
• Thus  is not provable from A, which is what 
claims, so  is true.
• Key:  is true, but we can’t prove it.
• We could always make A bigger, but then same
problem would arise in the new A for a new .
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General Logics
• Ontology:
– science or study of being, which problems are being
investigated, the kinds of abstract entities that are to be admitted
to a language system, things that exist in a system.
• Epistemology:
– study of methods and grounds of knowledge, its limits, and its
validity, theories of knowledge.
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Uncertainty
• The real world contains uncertainty, we
can never be sure of our premises and we
can never be sure our outcomes follow our
premises.
• Instead, we have degrees of belief about
the world. Our knowledge ideally reflects
degrees of belief in a way that helps us
make decisions when uncertainty
abounds.
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Why we need uncertainty
Let action At = leave for airport t minutes before flight
Will At get me there on time?
Problems:
1.
2.
2.
3.
3.
4.
4.
5.
partial observability (road state, other drivers' plans, etc.)
noisy sensors (traffic reports, road conditions)
uncertainty in action outcomes (flat tire, etc.)
immense complexity of modeling and predicting traffic
Hence a purely logical approach either
1.
2.
“A25 will
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risks falsehood: “A25 will get me there on time”, or
leads to conclusions that are too weak for decision making:
get me there on time if there's
no accident on the bridge, and it doesn't
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rain, and my tires remain intact, and they aren’t doing construction, etc
etc.”
Methods for handling uncertainty
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Default or nonmonotonic logic:
–
–
–
–
Assume my car does not have a flat tire
Assume A25 works unless contradicted by evidence
Key issue: we draw conclusions tentatively, reserving the right to retract a conclusion in light
of further information. Set of conclusions entailed by KB does not increase, and might
decrease, with size of the KB itself.
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Issues: What assumptions are reasonable? How to handle contradiction?
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Rules with fudge factors:
–
–
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–
–
A25 |→0.3 get there on time
Sprinkler |→ 0.99 WetGrass
WetGrass |→ 0.7 Rain
• Issues: Problems with combination, e.g., Sprinkler causes Rain??
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• Probability
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– Model agent's degree of belief
Probability as Belief
Probabilistic assertions summarize effects of
– laziness: failure to enumerate exceptions, qualifications, etc.
–
– ignorance: lack of relevant facts, initial conditions, etc.
But this presupposes a deterministic world, i.e., uncertainty exists only
from laziness or ignorance. Does randomness truly exist??
Subjective (Bayesian) probability:
• Probabilities relate propositions to agent's own state of knowledge
e.g., P(A25 | no reported accidents) = 0.06
These are not assertions about the world, rather agent’s belief.
These are NOT claims of a “probabilistic tendency” in current
situation (but might be learned from past experience of similar
situations)
Probabilities
of propositions change
with new evidence:
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e.g., P(A25 | no reported accidents, 5 a.m.) = 0.15
Making decisions under
uncertainty
Suppose I believe the following:
P(A25 gets me there on time | …)
P(A90 gets me there on time | …)
P(A120 gets me there on time | …)
P(A1440 gets me there on time | …)
= 0.04
= 0.70
= 0.95
= 0.9999
• Which action to choose?
•
Depends on my preferences for missing flight vs. time
spent waiting, etc.
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– Utility theory is used to represent and infer preferences
Probability basics
• Begin with set , sample space
– e.g., 6 possible rolls of a die
–  2  is a sample point, possible world, or atomic event
• A probability space (or prob. model) is a sample space
with a numeric assignment P() for every  2 , s.t.:
– 0 · P() · 1
–  P() = 1
• e.g., P(1)=P(2)=…=P(6) = 1/6
• An event A is any subset of 
– P(A) =  2 A P()
• E.g., P(die roll < 4) = P(1)+P(2)+P(3)+P(4)=3*1/6=1/2
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Random Variables
• A random variable is a function from sample points to
some range, e.g., the reals or booleans
– e.g., Odd(i) = true, whenever i is odd.
• P induces a probability distribution for any r.v. X:
– P(X=xi) = :X() = xi P()
• e.g., in die case, we have that P(Odd = true) =
P(1)+P(3)+P(5) = 1/2
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Propositions
• Think of a proposition as the event (set of basic sample
points in ) where the proposition is true.
• Given Boolean random variables A and B:
– event a = set of sample points where A() = true
– event :a = set of sample points where A() = false
– event aÆb = points where both A() = true and B() = truen
• Often in AI applications, sample points are defined by the
values of a set of random variables, i.e., sample space is
Cartesian product of r.v.’s ranges
• With boolean variables, sample point = model in
propositional logic (i.e., assignment to prop log vars)
– e.g., A=true, B=true, or aÆ: b
• Proposition = disjunction of atomic events that are true
– e.g., (a Ç b) ´ (: a Æ b) Ç (a Æ : b) Ç (a Æ b)
–  P(a Æ b) = P(: a Æ b) + P(a Æ : b) + P(a Æ b)
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Why use probability?
• The definitions imply that certain logically related
events must have related probabilities:
– P(A  B) = P(A) + P(B) - P(A  B)
– A rationality argument: probabilities are rational.
– de Finetti (1931): an agent who bets according to
probabilities that violate these axioms can be forced
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Syntax for propositions
• Propositional or Boolean random variables:
– E.g., Cavity (do I have a cavity?)
– Cavity=true is a proposition, also written just as cavity.
• Discrete random variables
•
e.g., Weather is one of <sunny,rainy,cloudy,snow>
Weather=train is a proposition
Domain values must be exhaustive and mutually exclusive
• Elementary proposition constructed by assignment of a value to a
• random variable: e.g., Weather = sunny, Cavity = false
• (abbreviated as cavity)
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• Complex
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propositions formed from
standard logical connectives e.g., Weather = sunny  Cavity = false
Syntax
• Atomic event: A complete specification of the
state of the world about which the agent is
uncertain
•
E.g., if the world consists of only two Boolean variables
Cavity and Toothache, then there are 4 distinct atomic
events:
Cavity = false Toothache = false
Cavity = false  Toothache = true
Cavity = true  Toothache = false
Cavity = true  Toothache = true
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• Atomic events are mutually
exclusive and
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Prior probability
•
•
Prior or unconditional probabilities of propositions
e.g., P(Cavity = true) = 0.1 and P(Weather = sunny) = 0.72 correspond to belief prior
to arrival of any (new) evidence
•
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Probability distribution gives values for all possible assignments:
P(Weather) = <0.72,0.1,0.08,0.1> (normalized, i.e., sums to 1)
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Joint probability distribution for a set of random variables gives the
probability of every atomic event on those random variables
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P(Weather,Cavity) = a 4 × 2 matrix of values:
Weather =
Cavity = true
Cavity = false
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sunny
0.144
0.576
rainy
0.02
0.08
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cloudy
0.016
0.064
snow
0.02
0.08
Conditional probability
•
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Conditional or posterior probabilities
e.g., P(cavity | toothache) = 0.8
i.e., given that toothache is all I know (at the moment). This is *NOT* saying that “if
toothache, then 80% chance of cavity”. There might be other information that
could either increase or decrease this probability that is simultaneously true
along with toothache.
•
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(Notation for conditional distributions:
P(Cavity | Toothache) = 2-element vector of 2-element vectors)
•
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If we know more, e.g., cavity is also given, then we have
P(cavity | toothache,cavity) = 1
Note: the less specific belief (toothache) remains valid after more evidence arrives,
but it is not always useful
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Conditional probability
• Definition of conditional probability:
•
P(a | b) = P(a  b) / P(b) if P(b) > 0
• Product rule gives an alternative formulation:
•
P(a  b) = P(a | b) P(b) = P(b | a) P(a)
• A general version holds for whole distributions, e.g.,
•
P(Weather,Cavity) = P(Weather | Cavity) P(Cavity)
• (View as a set of 4 × 2 equations, not matrix mult., rather “table
multiplication”)
•
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• Chain
•
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rule is derived by successive
Inference by enumeration
• Start with the joint probability distribution:
•
• For any proposition φ, sum the atomic events where it is
true: P(φ) = Σω:ω╞φ P(ω)
•
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Inference by enumeration
• Start with the joint probability distribution:
•
• For any proposition φ, sum the atomic events where it is
true: P(φ) = Σω:ω╞φ P(ω)
•
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• P(toothache) = 0.108 + 0.012 + 0.016 + 0.064 = 0.2
Inference by enumeration
• Start with the joint probability distribution:
•
• For any proposition φ, sum the atomic events where it is
true: P(φ) = Σω:ω╞φ P(ω)
•
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• P(toothache Ç cavity) = 0.108 + 0.012 + 0.016 + 0.064 =
Inference by enumeration
• Start with the joint probability distribution:
•
• Can also compute conditional probabilities:
•
P(cavity | toothache) = P(cavity  toothache)
P(toothache)
=
0.016+0.064
0.108 + 0.012 + 0.016 + 0.064
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0.4
Normalization
• Denominator can be viewed as a normalization constant α
•
P(Cavity | toothache) = α, P(Cavity,toothache)
= α, [P(Cavity,toothache,catch) + P(Cavity,toothache, catch)]
= α, [<0.108,0.016> + <0.012,0.064>]
= α, <0.12,0.08> = <0.6,0.4>
General idea: compute distribution on query variable by fixing evidence
variables and summing over hidden variables
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Inference by enumeration,
contd.
Let X be all the variables.
Typically, we are interested in
the posterior joint distribution of the query variables Y
given specific values e for the evidence variables E
Let the hidden variables be H = X - Y - E
Then the required summation of joint entries is done by summing out the hidden
variables:
P(Y | E = e) = αP(Y,E = e) = αΣhP(Y,E= e, H = h)
•
The terms in the summation are joint entries because Y, E and H together exhaust
the set of random variables
•
• Obvious problems:
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•
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1. Worst-case time complexity O(dn) where d is the largest arity
Independence
• A and B are independent iff
P(A|B) = P(A) or P(B|A) = P(B)
or P(A, B) = P(A) P(B)
P(Toothache, Catch, Cavity, Weather)
= P(Toothache, Catch, Cavity) P(Weather)
• 16 entries reduced to 10; for n independent biased coins, O(2n)
→O(n)
•
• Absolute independence powerful but rare
•
• Dentistry is a large field with hundreds of variables, none of which
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are independent. What to do? EE562
•
Conditional independence
• P(Toothache, Cavity, Catch) has 23 – 1 = 7 independent entries
•
• If I have a cavity, the probability that the probe catches in it doesn't
depend on whether I have a toothache:
•
(1) P(catch | toothache, cavity) = P(catch | cavity)
• The same independence holds if I haven't got a cavity:
•
(2) P(catch | toothache,cavity) = P(catch | cavity)
• Catch is conditionally independent of Toothache given Cavity:
•
P(Catch | Toothache,Cavity) = P(Catch | Cavity)
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• Equivalent statements:
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Conditional independence
contd.
• Write out full joint distribution using chain rule:
•
P(Toothache, Catch, Cavity)
= P(Toothache | Catch, Cavity) P(Catch, Cavity)
= P(Toothache | Catch, Cavity) P(Catch | Cavity) P(Cavity)
= P(Toothache | Cavity) P(Catch | Cavity) P(Cavity)
I.e., 2 + 2 + 1 = 5 independent numbers
• In most cases, the use of conditional independence
reduces the size of the representation of the joint
distribution from exponential
in n to linear in n.
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•
Bayes' Rule
• Product rule P(ab) = P(a | b) P(b) = P(b | a) P(a)
•
 Bayes' rule: P(a | b) = P(b | a) P(a) / P(b)
• or in distribution form
•
P(Y|X) = P(X|Y) P(Y) / P(X) = αP(X|Y) P(Y)
• Useful for assessing diagnostic probability from causal
probability:
•
– P(Cause|Effect) = P(Effect|Cause) P(Cause) / P(Effect)
–
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– E.g., let M be meningitis, S be stiff neck:
Bayes' Rule and conditional
independence
P(Cavity | toothache  catch)
= αP(toothache  catch | Cavity) P(Cavity)
= αP(toothache | Cavity) P(catch | Cavity) P(Cavity)
• This is an example of a naïve Bayes model:
•
P(Cause,Effect1, … ,Effectn) = P(Cause) πiP(Effecti|Cause)
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Key Benefit
• Probabilistic reasoning (using things like
conditional probability, conditional
independence, and Bayes’ rule) make it
possible to make reasonable decisions
amongst a set of actions, that otherwise
(without probability, as in propositional or
first order logic) we would have to resort to
random guessing.
• Example: Wumpus World
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Summary
• Probability is a rigorous formalism for uncertain
knowledge
•
• Joint probability distribution specifies probability
of every atomic event
• Queries can be answered by summing over
atomic events
•
• For nontrivial domains, we must find a way to
reduce the joint size
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• Independence and conditional independence