1. 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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2. Uncertainty
Chapter 13
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3. Outline Limitations of logic (Gödel) Uncertainty Probability
Syntax and Semantics
Inference
Independence and Bayes' Rule
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4. Reminder
HW4 Due Today!!
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5. General Logics Ontology: Epistemology:
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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6. 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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7. 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 …
Enemy(Nono,America)
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8. 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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9. 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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10. 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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11. 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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12. 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
In A, assume  false, then it is true.
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13. 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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14. General Logics Ontology: Epistemology:
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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15. 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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16. Why we need uncertainty
Let action At = leave for airport t minutes before flight
Will At get me there on time?
Problems:
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
risks falsehood: “A25 will get me there on time”, or
leads to conclusions that are too weak for decision making:
“A25 will get me there on time if there's no accident on the bridge, and it doesn't rain, and my tires remain intact, and they aren’t doing construction, etc etc.”
(A1440 might reasonably be said to get me there on time but I'd have to stay overnight in the airport …)
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17. Methods for handling uncertainty
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.
Issues: What assumptions are reasonable? How to handle contradiction?
Rules with fudge factors:
A25 |→0.3 get there on time
Sprinkler |→ 0.99 WetGrass
WetGrass |→ 0.7 Rain
Issues: Problems with combination, e.g., Sprinkler causes Rain??
Probability
Model agent's degree of belief
Given the available evidence,
A25 will get me there on time with probability 0.04
Fuzzy logic handles degree of truth, NOT uncertainty.
E.g., WetGrass is true to degree 0.2
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18. 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:
e.g., P(A25 | no reported accidents, 5 a.m.) = 0.15
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19. Making decisions under uncertainty
Suppose I believe the following:
P(A25 gets me there on time | …) = 0.04
P(A90 gets me there on time | …) = 0.70
P(A120 gets me there on time | …) = 0.95
P(A1440 gets me there on time | …) =
Which action to choose?
Depends on my preferences for missing flight vs. time spent waiting, etc.
Utility theory is used to represent and infer preferences
Decision theory = probability theory + utility theory
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20. 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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21. 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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22. 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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23. 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 to bet so as to loose money regardless of outcome.
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24. 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
(abbreviated as cavity)
Complex propositions formed from elementary propositions and standard logical connectives e.g., Weather = sunny  Cavity = false
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25. 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
Atomic events are mutually exclusive and exhaustive
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26. 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
Probability distribution gives values for all possible assignments:
P(Weather) = <0.72,0.1,0.08,0.1> (normalized, i.e., sums to 1)
Joint probability distribution for a set of random variables gives the probability of every atomic event on those random variables
P(Weather,Cavity) = a 4 × 2 matrix of values:
Weather = sunny rainy cloudy snow
Cavity = true
Cavity = false
Every question about a domain can be answered by the joint distribution
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27. Conditional probability
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.
(Notation for conditional distributions:
P(Cavity | Toothache) = 2-element vector of 2-element vectors)
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
New evidence may be irrelevant, allowing simplification, e.g.,
P(cavity | toothache, SeaHawksWin) = P(cavity | toothache) = 0.8
This kind of inference, sanctioned by domain knowledge, is crucial
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28. 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”)
Chain rule is derived by successive application of product rule:
P(X1, …,Xn) = P(X1,...,Xn-1) P(Xn | X1,...,Xn-1)
= P(X1,...,Xn-2) P(Xn-1 | X1,...,Xn-2) P(Xn | X1,...,Xn-1)
= …
= πi= 1^n P(Xi | X1, … ,Xi-1)
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29. 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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30. 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)
P(toothache) = = 0.2
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31. 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)
P(toothache Ç cavity) = = 0.2
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32. Inference by enumeration
Start with the joint probability distribution:
Can also compute conditional probabilities:
P(cavity | toothache) = P(cavity  toothache)
P(toothache) =
= 0.4
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33. 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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34. 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:
Worst-case time complexity O(dn) where d is the largest arity
Space complexity O(dn) to store the joint distribution
How to find the numbers for O(dn) entries?
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35. 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 are independent. What to do?
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36. 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)
Equivalent statements:
P(Toothache | Catch, Cavity) = P(Toothache | Cavity)
P(Toothache, Catch | Cavity) = P(Toothache | Cavity) P(Catch | Cavity)
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37. 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., = 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.
Conditional independence is our most basic and robust form of knowledge about uncertain environments.
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38. 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)
E.g., let M be meningitis, S be stiff neck:
P(m|s) = P(s|m) P(m) / P(s) = 0.8 × / 0.1 =
Note: posterior probability of meningitis still very small!
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39. 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)
Total number of parameters is linear in n
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40. 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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41. 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
Independence and conditional independence provide the tools
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