Exact Inference CSci 5512: Artificial Intelligence II

  Instructor: Paul Schrater

     

Overview: Inference Tasks

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

Overview: Inference Tasks

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

Overview: Inference Tasks

  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)

Overview: Inference Tasks

  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?

Overview: Inference Tasks

  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)

Inference by Enumeration (Contd)

  Complexity can be improved by a simple observation

P(b|j,m) =   1 P(j,m) E A

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

=   1 P(j,m) E A

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

=   1 P(j,m)

P(b) E

P(E) A

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

Complexity is O(2n)   Some computations are repeated

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�

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