Artificial Intelligence CS 165A Tuesday, November 20, 2007 Knowledge Representation (Ch 10) Uncertainty (Ch 13)

Artificial Intelligence

CS 165A

Tuesday, November 20, 2007

Knowledge Representation (Ch 10) Uncertainty (Ch 13)

2

Notes

• HW #4 due by noon tomorrow

• Reminder: Final exam December 14, 4-7pm– Review in class on Dec. 6th

3

Situation Calculus – actions, events

• “Situation Calculus” is a way of describing change over time in first-order logic– Fluents: Functions or predicates that can vary over time have an

extra argument, Si (the situation argument)

Predicate(args, Si) Location of an agent, aliveness, changing properties, ...

– The Result function is used to represent change from one situation to another resulting from an action (or action sequence)

Result(GoForward, Si) = Sj

“Sj is the situation that results from the action GoForward applied to situation Si

Result() indicates the relationship between situations

Review

4

Situation Calculus

Represents the world in different “situations” and the relationship between situations

Review

5

Situation Calculus

Represents the world in different “situations” and the relationship between situations

Review

6

Examples

• How would you interpret the following sentences in First-Order Logic using situation calculus?

x, s Studying(x, s) Failed(x, Result(TakeTest, s))

x, s TurnedOn(x, s) LightSwitch(x) TurnedOff(x, Result(FlipSwitch, s))

Review

If you’re studying and then you take the test, you will fail.

(or) Studying a subject implies that you will fail the test for that subject.

If you flip the light switch when it is turned on, it will then be turned off.

7

There are other ways to deal with time

• Event calculus– Based on points in time rather than situations– Designed to allow reasoning over periods of time

Can represent actions with duration, overlapping actions, etc.

• Generalized events– Parts of a general “space-time chunk”

• Processes– Not just discrete events

• Intervals– Moments and durations of time

• Objects with state fluents– Not just events, but objects can also have time properties

8

Event calculus relations

• Initiates(e, f, t)– Event e at time t causes fluent f to become true

• Terminates(e, f, t)– Event e at time t causes fluent f to no longer be true

• Happens(e, t)– Event e happens at time t

• Clipped(f, t1, t2)

– f is terminated by some event sometime between t1 and t2

9

Generalized events

• An ontology of time that allows for reasoning about various temporal events, subevents, durations, processes, intervals, etc.

time

Australia

Space-time chunk

10

Time interval predicates

After(ReignOf(ElizabethII), ReignOf(GeorgeVI))

Overlap(Fifties, ReignOf(Elvis))

Start(Fifties) = Start(AD1950)

Meet(Fifties, Sixties)

Ex:

11

Objects with state fluents

President(USA)

12

Knowledge representation

• Chapter 10 covers many topics in knowledge representation, many of which are important to real, sophisticated AI reasoning systems– We’re only scratching the surface of this topic

– Best covered in depth in an advanced AI course and in context of particular AI problems

– Read through the Internet shopping world example in 10.5

• Now we move on to probabilistic reasoning, a different way of representing and manipulating knowledge– Chapters 13 and 14

13

Quick Review of Probability

From here on we will assume that you know this…

14

Probability notation and notes

• Probabilities of propositions

– P(A), P(the sun is shining)

• Probabilities of random variables

– P(X = x1), P(Y = y1), P(x1 < X < x2)

• P(A) usually means P(A = True) (A is a proposition, not a variable)

– This is a probability value

– Technically, P(A) is a probability function

• P(X = x1)

– This is a probability value (P(X) is a probability function)

• P(X)– This is a probability function or a probability density function

• Technically, if X is a variable, we should not write P(X) = 0.5

– But rather P(X = x1) = 0.5

15

Discrete and continuous probabilities

• Discrete: Probability function P(X, Y) is described by an MxN matrix of probabilities– Possible values of each: P(X=x1, Y=y1) = p1

– P(X=xi, Y=yj) = 1

– P(X, Y, Z) is an MxNxP matrix

• Continuous: Probability density function (pdf) P(X, Y) is described by a 2D function– P(x1 < X < x2, y1 < Y < y2) = p1

– P(X, Y) dX dY = 1

16

Discrete probability distribution

0

0.1

0.2

1 2 3 4 5 6 7 8 9 10 11 12

X

p(X)

1)( i

ixXp

17

Continuous probability distribution

0

0.2

0.4

1 2 3 4 5 6 7 8 9 10 11 12

X

p(X)

1)(

Xp

18

Continuous probability distribution

0

0.2

0.4

1 2 3 4 5 6 7 8 9 10 11 12

X

p(X)

aXp 8

6

)(P(X=5) = 0

P(X=x1) = 0

P(X=5) = ???

19

Three Axioms of Probability

1. The probability of every event must be nonnegative– For any event A, P(A) 0

2. Valid propositions have probability 1– P(True) = 1

– P(A A) = 1

3. For disjoint events A1, A2, …

– P(A1 A2 …) = P(A1) + P(A2) + …

• From these axioms, all other properties of probabilities can be derived.– E.g., derive P(A) + P(A) = 1

20

Some consequences of the axioms

• Unsatisfiable propositions have probability 0– P(False) = 0

– P(A A) = 0

• For any two events A and B– P(A B) = P(A) + P(B) – P(A B)

• For the complement Ac of event A– P(Ac) = 1 – P(A)

• For any event A– 0 P(A) 1

• For independent events A and B– P(A B) = P(A) P(B)

21

Venn Diagram

True

A BA B

Visualize: P(True), P(False), P(A), P(B), P(A), P(B),P(A B), P(A B), P(A B), …

22

Joint Probabilities

• A complete probability model is a single joint probability distribution over all propositions/variables in the domain

– P(X1, X2, …, Xi, …)

• A particular instance of the world has the probability

– P(X1=x1 X2=x2 … Xi=xi …) = p

• Rather than stating knowledge as– Raining WetGrass

• We can state it as– P(Raining, WetGrass) = 0.15

– P(Raining, WetGrass) = 0.01

– P(Raining, WetGrass) = 0.04

– P(Raining, WetGrass) = 0.8

0.8 0.04

0.01 0.15

WetGrass WetGrass

Raining

Raining

23

Conditional Probability

• Unconditional, or Prior, Probability– Probabilities associated with a proposition or variable, prior to any

evidence

– E.g., P(WetGrass), P(Raining)

• Conditional, or Posterior, Probability– Probabilities after evidence is gathered

– P(A | B) – “The probability of A given that we know B”

– After (posterior to) procuring evidence

– E.g., P(WetGrass | Raining)

)(

),()|(

YP

YXPYXP ),()()|( YXPYPYXP or

Assumes P(Y) nonzero

24

The chain rule

)()|(),( YPYXPYXP

),|()|()(

,

)()|(),|(

),(),|(),,(

YXZPXYPXP

lyequivalentor

ZPZYPZYXP

ZYPZYXPZYXP

By the Chain Rule

• Precedence: ‘|’ is lowest• E.g., P(X | Y, Z) means which?

P( (X | Y), Z )P(X | (Y, Z) )

Notes:

25

Joint probability distribution

From P(X,Y), we can always calculate: P(X)P(Y)P(X|Y)P(Y|X)

P(X=x1)

P(Y=y2)

P(X|Y=y1)

P(Y|X=x1)

P(X=x1|Y)etc.

0.30.20.1

0.10.10.2

X

Y

x1 x2 x3

y1

y2

26

0.2 0.1 0.1

0.1 0.2 0.3

x1 x2 x3

y1

y2

P(X,Y)

0.40.30.3

x1 x2 x3

P(X)

0.6

0.4y1

y2

P(Y)

0.50.3330.167

0.250.250.5

x1 x2 x3

y1

y2

P(X|Y)

0.750.6670.333

0.250.3330.667

x1 x2 x3

y1

y2

P(Y|X)P(X=x1,Y=y2) = ?

P(X=x1) = ?

P(Y=y2) = ?

P(X|Y=y1) = ?

P(X=x1|Y) = ?

27

Probability Distributions

Continuous vars

Scalar* Scalar

Function of two variables MxN matrix

Function of two variables MxN matrix

Function of one variable M vector

Function of one variable N vector

Scalar* Scalar

Discrete vars

Function (of one variable) M vector

P(X=x)

P(X,Y)

P(X|Y)

P(X|Y=y)

P(X=x|Y)

P(X=x|Y=y)

P(X)

* - actually zero. Should be P(x1 < X < x2)

28

Bayes’ Rule

• Since

and

• Then

)()|(),( YPYXPYXP

)()|(),( XPXYPYXP

)(

)()|()|(

YP

XPXYPYXP Bayes’ Rule

)()|()()|( XPXYPYPYXP

29

Bayes’ Rule

• Similarly, P(X) conditioned on two variables:

),,|(

),,|(),,,|(),,,|(

32

31312321

N

NNN XXXP

XXXPXXXXPXXXXP

)|(

)|(),|(),|(

ZYP

ZXPZXYPZYXP

• Or N variables:

)|(

)|(),|(),|(

YZP

YXPYXZPZYXP

30

Bayes’ Rule

)(

)()|()|(

DP

HPHDPDHP ii

i Posterior probability

(diagnostic knowledge)

Likelihood(causal knowledge) Prior probability

• This simple equation is very useful in practice– Usually framed in terms of hypotheses (H) and data (D)

Which of the hypotheses is best supported by the data?

Normalizing constant

)()|()|( iii HPHDPkDHP

31

Bayes’ rule example: Medical diagnosis

• Meningitis causes a stiff neck 50% of the time

• A patient comes in with a stiff neck – what is the probability that he has meningitis?

• Need to know two things:– The prior probability of a patient having meningitis (1/50,000)

– The prior probability of a patient having a stiff neck (1/20)

• ?

• P(M | S) = (0.5)(0.00002)/(0.05) = 0.0002

)(

)()|()|(

SP

MPMSPSMP

32

Example (cont.)

• Suppose that we also know about whiplash– P(W) = 1/1000

– P(S | W) = 0.8

• What is the relative likelihood of whiplash and meningitis?– P(W | S) / P(M | S)

016.005.0

)001.0)(8.0(

)(

)()|()|(

SP

WPWSPSWP

So the relative likelihood of whiplash vs. meningitis is (0.016/0.0002) = 80

33

A useful Bayes rule example

A test for a new, deadly strain of anthrax (that has no symptoms) is known to be 99.9% accurate. Should you get tested? The chances of having this strain are one in a million.

What are the random variables?A – you have anthrax (boolean)

T – you test positive for anthrax (boolean)

Notation: Instead of P(A=True) and P(A=False), we will write P(A) and P(A)

What do we want to compute?P(A|T)

What else do we need to know or assume?Priors: P(A) , P(A)

Given: P(T|A) , P(T|A), P(T|A), P(T|A)

A

T

A

TA

T

A

T

Possibilities

34

Example (cont.)

We know:Given: P(T|A) = 0.999, P(T|A) = 0.001, P(T|A) = 0.001, P(T|A)

= 0.999

Prior knowledge: P(A) = 10-6, P(A) = 1 – 10-6

Want to know P(A|T)P(A|T) = P(T|A) P(A) / P(T)

Calculate P(T) by marginalizationP(T) = P(T|A) P(A) + P(T|A) P(A) = (0.999)(10-6) + (0.001)(1 – 10-6)

0.001

So P(A|T) = (0.999)(10-6) / 0.001 0.001

Therefore P(A|T) 0.999

What if you work at a Post Office?

35All people

People without anthraxPeople with anthrax

Good T

Bad T(0.1%)