Exact Inference - University of...

Exact Inference CSci 5512: Artificial Intelligence II

Instructor: Paul Schrater

Overview: Inference Tasks

Simple Queries: Compute posterior marginals P(b|j,¬m)


Simple Queries: Compute posterior marginals P(b|j,¬m) Conjunctive Queries: Compute joint marginals P(b,¬e|j,¬m)


Simple Queries: Compute posterior marginals P(b|j,¬m) Conjunctive Queries: Compute joint marginals P(b,¬e|j,¬m) Optimal Decisions: Compute P(outcome|action,evidence)


Simple Queries: Compute posterior marginals P(b|j,¬m) Conjunctive Queries: Compute joint marginals P(b,¬e|j,¬m) Optimal Decisions: Compute P(outcome|action,evidence) Value of Information: Which evidence to seek next?


Simple Queries: Compute posterior marginals P(b|j,¬m) Conjunctive Queries: Compute joint marginals P(b,¬e|j,¬m) Optimal Decisions: Compute P(outcome|action,evidence) Value of Information: Which evidence to seek next? Explanation: Why do I need a new starter motor?

Inference

.001 P(B)

.002 P(E)

Earthquake

MaryCalls JohnCalls

B T T F F

E T F T F

.95

.94

.29

.001

Burglary

P(A|B,E)

T F

.90

.05

Alarm

A P(J|A) T F

.70

.01

A P(M|A)

How can we compute P(b|j,m)?

Inference by Enumeration

Simple query can be answered using Bayes rule

Instructor: Arindam Banerjee Exact Inference

Inference by Enumeration

Simple query can be answered using Bayes rule From Bayes Rule

P(b|j,m) = P(b,j,m) P(j,m)

Each marginal can be obtained from the joint distribution

E A P(b,j,m)

P(j,m)

=

= B E A

P(b,E,A,j,m)

P(B,E,A,j,m)

Each term can be written as product of conditionals

P(b,E,A,j,m) = P(b)P(E)P(A|b,E)P(j|A)P(m|A)

The complexity of the simple approach is O(n2n)

Burglary Network

.001

P(B)

.002

P(E) Earthquake

MaryCalls JohnCalls

B T T F F

E T F T F

.95

.94

.29

.001

Burglary

P(A|B,E)

T F

.90

.05

Alarm

A P(J|A) T F

.70

.01

A P(M|A)

P(j|a) .90

P(m|a) .70

P(j| a) P(j|a) .05

P(m| a) .01

P(j| a)

Enumeration Tree

P(e) .002

P( e) .998

P(a|b,e) .95

P( a|b, e) .06

P( a|b,e) .05

P(a|b, e) .94

.05

P(m| a) .01

Instructor: Arindam Banerjee

P(j|a) .90

P(m|a) .70

.05

P(m| a) .01

P(j| a) P(j|a) .90

P(m|a) .70

.05

P(m| a) .01

P(j| a)

Enumeration Tree

P(b) .001

P(e) .002

P( e) .998

P(a|b,e) .95

P( a|b, e) .06

P( a|b,e) .05

P(a|b, e) .94

Enumeration Tree for P(b|j,m) P(j|a)P(m|a) and P(j|¬a)P(m|¬a) are computed twice

Variable Elimination

Queries can be written as sum-of-products Main idea Sum over products to eliminate variables Storage required to prevent repeat computations Burglary network

P(b|j,m)

= 1 P(j,m)

P(b) B E

P(E) E A

P(A|b,E)P(j|A)P(m|A) A J M

Have a factor for every variable

Factors for Variable Elimination

Factor for M: fm(A) = [P(m|a) P(m|¬a)] Factor for J: fj(A) = [P(j|a) P(j|¬a)] Similarly, factors fA(B,E,A),fE(E),fB(B) Summing out to eliminate variables

¯

¯ ¯

fAjm(B,E) =

fEAjm(B) = A

E ¯

fA(B,E,A)fj(A)fm(A)

fE(E)fAjm(B,E)

P(B|j,m) = 1 P(j,m)

¯ ¯ fB(B)fEAjm(B)

Variable Elimination: Basic Operations

Summing out a variable from a product of factors Pointwise products of (a pair of) factors Sum out a variable from a product of factors Pointwise product of factors g1(a,b) × g2(b,c) = g(a,b,c) In general g1(x1,...,xj,y1,...,yk) × g2(y1,...,yk,z1,...,zl) = g(x1,...,xj,y1,...,yk,z1,...,zl)

Assuming f1,...,fi do not depend on X

x f1 × ··· × fk

X

fi+1 × ··· × fk = f1 × ··· × fi ×

= f1 × ··· × fi × fX¯

Example: Pointwise Product of Factors

A T T F F

B T F T F

f1(A,B) 0.3 0.7 0.9 0.1

B T T F F

C T F T F

f2(B,C) 0.2 0.8 0.6 0.4

A T T T T

B T T F F

C T F T F

f3(A,B,C) 0.3 × 0.2 0.3 × 0.8 0.7 × 0.6 0.7 × 0.4

T T F F

T F T F

0.9 × 0.2 0.9 × 0.8 0.1 × 0.6 0.1 × 0.4

Algorithm�

Irrelevant variables�

Complexity of Exact Inference

Singly connected networks (or polytrees) Any two nodes are connected by at most one path Time and space cost of variable elimination are O(dkn)

Multiply connected networks Can reduc 3SAT to exact inference =⇒ NP-hard Equivalent to counting 3SAT models =⇒ #P-complete

Inference Problems: Big Picture

Broadly the following types of problems Compute Compute Compute Compute

likelihood of observations marginals P(xA) on subset A of nodes conditionals P(xA|xB) on disjoint subsets A,B mode argmaxx P(x)

• First 3 problems are similar Involves marginalization over sum-of-products • Last problem is fundamentally different Entails maximization rather than marginalization • There is an important connection between the problems

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