Lecture 16 • 1 Lecture 16 • 2 · 1 Lecture 16 • 1 6.825 Techniques in Artificial Intelligence...

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Lecture 16 • 1

6.825 Techniques in Artificial Intelligence

Inference in Bayesian Networks

• Exact inference• Approximate inference

Lecture 16 • 2

Query Types

Given a Bayesian network, what questions might wewant to ask?

• Conditional probability query: P(x | e)• Maximum a posteriori probability:

What value of x maximizes P(x|e) ?

General question: What’s the whole probabilitydistribution over variable X given evidence e,P(X | e)?

Lecture 16 • 3

Using the joint distribution

To answer any query involving a conjunction ofvariables, sum over the variables not involved inthe query.

†

Pr(d) = Pr(a,b,c,d)ABCÂ

= Pr(A = a Ÿ B = b ŸC = c)cŒdom(C )Â

bŒdom(B )Â

aŒdom(A )Â

†

Pr(d | b) =Pr(b,d)Pr(b)

=Pr(a,b,c,d)

ACÂ

Pr(a,b,c,d)ACDÂ

Lecture 16 • 4

Simple Case

A B C D

†

Pr(d) = Pr(a,b,c,d)ABCÂ

= Pr(d | c)Pr(c | b)Pr(b | a)Pr(a)ABCÂ

= Pr(d | c)Pr(c | b)Pr(b | a)Pr(a)AÂ

BÂ

CÂ

= Pr(d | c) Pr(c | b) Pr(b | a)Pr(a)AÂ

BÂ

CÂ

Lecture 16 • 5

Simple Case

A B C D

†

Pr(d) = Pr(d | c) Pr(c | b) Pr(b | a)Pr(a)AÂ

BÂ

CÂ

†

Pr(b1 | a1) Pr(a1) Pr(b1 | a2 ) Pr(a2 )Pr(b2 | a1) Pr(a1 ) Pr(b2 | a2 ) Pr(a2 )

È

Î Í

˘

˚ ˙

Lecture 16 • 6

Simple Case

A B C D

†


BÂ

CÂ

†

Pr(b1 | a) Pr(a)A

ÂPr(b2 | a) Pr(a)

AÂ

È

Î

Í Í Í

˘

˚

˙ ˙ ˙

2

Lecture 16 • 7

Simple Case

A B C D

†


BÂ

CÂ

†

f1(b)

Lecture 16 • 8

Simple Case

B C D

†

Pr(d) = Pr(d | c) Pr(c | b) f1BÂ

CÂ (b)

†

f2 (c)

Lecture 16 • 9

Simple Case

B C D

†

Pr(d) = Pr(d | c) Pr(c | b) f1BÂ

CÂ (b)

†

f2 (c)

Lecture 16 • 10

Simple Case

C D

†

Pr(d) = Pr(d | c) f2 (c)CÂ

Lecture 16 • 11

Variable Elimination Algorithm

Given a Bayesian network, and an elimination orderfor the non-query variables, compute

For i = m downto 1• remove all the factors that mention Xi

• multiply those factors, getting a value for eachcombination of mentioned variables

• sum over Xi

• put this new factor into the factor set

†

K Pr(x j | Pa(x jj

’Xm

ÂX2

ÂX1

Â ))

Lecture 16 • 12

One more example

Smoke

Bronchitis

Lungcancer

DyspneaXray

chest

Abnrm

Tuberculosis

risky

Visit

†

Pr(d) =Pr(d | a,b) Pr(a | t,l ) Pr(b | s) Pr(l | s) Pr(s)Pr(x | a) Pr(t | v) Pr(v)A,B ,L,T,S ,X ,V

Â

4

Lecture 16 • 19

One more example

Bronchitis

Lungcancer

Dyspnea

chest

Abnrm

Tuberculosis

†

Pr(d) = Pr(d | a,b) f2 (b,l) Pr(a | t,l) f1(t)TÂ

A,B ,LÂ

†

f3 (a,l )

Lecture 16 • 20

One more example

Bronchitis

Lungcancer

Dyspnea

chest

Abnrm

†

Pr(d) = Pr(d | a,b) f2 (b,l) f3(a,l )A,B ,LÂ

Lecture 16 • 21

One more example

Bronchitis

Lungcancer

Dyspnea

chest

Abnrm

†

Pr(d) = Pr(d | a,b) f2 (b,l) f3 (a,l )L

ÂA ,BÂ

†

f4 (a,b)

Lecture 16 • 22

One more example

Bronchitis

Dyspnea

chest

Abnrm

†

Pr(d) = Pr(d | a,b) f4 (a,b)A ,BÂ

Lecture 16 • 23

One more example

Bronchitis

Dyspnea

chest

Abnrm

†

Pr(d) = Pr(d | a,b) f4 (a,b)BÂ

AÂ

†

f5(a)Lecture 16 • 24

One more example

chest

Abnrm

†

Pr(d) = f5(a)AÂ

5

Lecture 16 • 25

Properties of Variable Elimination

• Time is exponential in size of largest factor• Bad elimination order can generate huge factors• NP Hard to find the best elimination order• Even the best elimination order may generate large

factors• There are reasonable heuristics for picking an

elimination order (such as choosing the variablethat results in the smallest next factor)

• Inference in polytrees (nets with no cycles) is linearin size of the network (the largest CPT)

• Many problems with very large nets have only smallfactors, and thus efficient inference

Lecture 16 • 26

Sampling

To get approximate answer we can do stochastic simulation(sampling).

A

B C

D

P(A) = 0.4

…

TTFT

DCBA

•Flip a coin where P(T)=0.4, assumewe get T, use that value for A

•Given A=T, lookup P(B|A=T) and flipa coin with that prob., assume we getF

•Similarly for C and D

•We get one sample from jointdistribution of these four vars

Estimate:

P*(D|A) = #D,A / #A

Lecture 16 • 27

Estimation

• Some probabilities are easier than others toestimate

• In generating the table, the rare events will not bewell represented

• P(Disease| spots-on-your-tongue, sore toe)• If spots-on-your-tongue and sore toe are not root

nodes, you would generate a huge table but thecases of interest would be very sparse in the table

• Importance sampling lets you focus on the set ofcases that are important to answering yourquestion

Lecture 16 • 28

Recitation Problem

• Do the variable elimination algorithm on the netbelow using the elimination order A,B,C (that is,eliminate node C first). In computing P(D=d), whatfactors do you get?

• What if you wanted to compute the whole marginaldistribution P(D)?

A B C D

Lecture 16 • 29

Another Recitation Problem

A

IHGFEDCB

MLKJ

Find an elimination order that keeps the factors smallfor the net below, or show that there is no suchorder.

N O

P

Lecture 16 • 30

The Last Recitation Problem (in thislecture)

Bayesian networks (or related models) are oftenused in computer vision, but they almost alwaysrequire sampling. What happens when you try todo variable elimination on a model like the gridbelow?

A B C D E

P Q R S T

K L M N O

F G H I J

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