EE562ARTIFICIAL INTELLIGENCE FOR ENGINEERSLecture 14, 5/23/2005

University of Washington, Department of Electrical EngineeringSpring 2005Instructor: Professor Jeff A. Bilmes

UncertaintyChapter 13

OutlineLimitations of logic (Gdel)UncertaintyProbabilitySyntax and SemanticsInferenceIndependence and Bayes' Rule

ReminderHW4 Due Today!!

General LogicsOntology: 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.

Example knowledge baseThe 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

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 WestMissile(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)

Forward chaining proofNote: can be seen as simply a list of logic sentences. So, any proof isjust a other facts that can be deduced from the KB to achieve new knowledge.

Gdels 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.

Gdels Incompleteness Theorem (general idea)Let #() be the number of sentence Let #-1(j) be the sentence for number jNote 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.

Gdels 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.

Gdels 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 falseIn A, assume false, then it is true.

Gdels Incompleteness Theorem (basic idea)Assume follows from A, but then is false (since says it doesnt 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 cant prove it.We could always make A bigger, but then same problem would arise in the new A for a new .

General LogicsOntology: 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.

UncertaintyThe 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.

Why we need uncertaintyLet action At = leave for airport t minutes before flightWill 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 eitherrisks falsehood: A25 will get me there on time, orleads 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 arent 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 )

Methods for handling uncertaintyDefault or nonmonotonic logic:

Assume my car does not have a flat tire

Assume A25 works unless contradicted by evidenceKey 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 RainIssues: 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 0.04Fuzzy logic handles degree of truth, NOT uncertainty. E.g., WetGrass is true to degree 0.2

Probability as BeliefProbabilistic 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 knowledgee.g., P(A25 | no reported accidents) = 0.06

These are not assertions about the world, rather agents 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

Making decisions under uncertaintySuppose 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 | ) = 0.9999

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

Probability basicsBegin with set , sample spacee.g., 6 possible rolls of a die 2 is a sample point, possible world, or atomic eventA probability space (or prob. model) is a sample space with a numeric assignment P() for every 2 , s.t.:0 P() 1 P() = 1e.g., P(1)=P(2)==P(6) = 1/6An 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

Random VariablesA random variable is a function from sample points to some range, e.g., the reals or booleanse.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

PropositionsThink 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() = trueevent :a = set of sample points where A() = falseevent ab = points where both A() = true and B() = truenOften 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 rangesWith boolean variables, sample point = model in propositional logic (i.e., assignment to prop log vars)e.g., A=true, B=true, or a: bProposition = disjunction of atomic events that are truee.g., (a b) (: a b) (a : b) (a b) P(a b) = P(: a b) + P(a : b) + P(a b)

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.

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 Weather=train is a propositionDomain 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)

Complex propositions formed from elementary propositions and standard logical connectives e.g., Weather = sunny Cavity = false

SyntaxAtomic 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 = falseCavity = false Toothache = trueCavity = true Toothache = falseCavity = true Toothache = true

Atomic events are mutually exclusive and exhaustive

Prior probabilityPrior 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) = (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 =sunnyrainycloudysnow Cavity = true 0.1440.02 0.016 0.02Cavity = false0.5760.08 0.064 0.08

Every question about a domain can be answered by the joint distribution

Conditional probabilityConditional 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) = 1Note: 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) = 0.8This kind of inference, sanctioned by domain knowledge, is crucial

Conditional probabilityDefinition 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) = = i= 1^n P(Xi | X1, ,Xi-1)

Inference by enumerationStart with the joint probability distribution:

For any proposition , sum the atomic events where it is true: P() = : P()

Inference by enumerationStart with the joint probability distribution:

For any proposition , sum the atomic events where it is true: P() = : P()

P(toothache) = 0.108 + 0.012 + 0.016 + 0.064 = 0.2

Inference by enumerationStart with the joint probability distribution:

For any proposition , sum the atomic events where it is true: P() = : P()

P(toothache cavity) = 0.108 + 0.012 + 0.016 + 0.064 = 0.2

Inference by enumerationStart 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= 0.4

NormalizationDenominator can be viewed as a normalization constant

P(Cavity | toothache) = , P(Cavity,toothache) = , [P(Cavity,toothache,catch) + P(Cavity,toothache, catch)]= , [ + ] = , =

General idea: compute distribution on query variable by fixing evidence variables and summing over hidden variables

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) = P(Y,E = e) = hP(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?

IndependenceA and B are independent iffP(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?

Conditional independenceP(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)

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.

Conditional independence is our most basic and robust form of knowledge about uncertain environments.

Bayes' RuleProduct rule P(ab) = 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) = P(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 0.0001 / 0.1 = 0.0008

Note: posterior probability of meningitis still very small!

Bayes' Rule and conditional independenceP(Cavity | toothache catch) = P(toothache catch | Cavity) P(Cavity) = P(toothache | Cavity) P(catch | Cavity) P(Cavity)

This is an example of a nave Bayes model:

P(Cause,Effect1, ,Effectn) = P(Cause) iP(Effecti|Cause)

Total number of parameters is linear in n

Key BenefitProbabilistic 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

SummaryProbability is a rigorous formalism for uncertain knowledge

Joint probability distribution specifies probability of every atomic eventQueries 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