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• Due Mon 4 Nov at 11:59pm
• Offi

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CS 4100: Artificial Intelligence Bayes’ Nets: Inference

Jan-Willem van de Meent, Northeastern University

[These slides were created by Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC Berkeley. All CS188 materials are available at http://ai.berkeley.edu.]

Bayes’ Net Representation

• A

A di directed, d, acyclic graph ph, o , one n node p per r random v variable

• A

A co conditional al probab ability tab able (CP CPT) ) for each node de

• A collection of distributions over X, one for

each possible assignment to parent variables

• Ba

Bayes’ne nets implicitly enc ncode jo join int dis istrib ributio ions

• As a product of local conditional distributions
• To see what probability a BN gives to a full assignment,

multiply all the relevant conditionals together:

Example: Alarm Network

B P(B) +b 0.001

• b

0.999 E P(E) +e 0.002

• e

0.998 B E A P(A|B,E) +b +e +a 0.95 +b +e

• a

0.05 +b

• e

+a 0.94 +b

• e
• a

0.06

• b

+e +a 0.29

• b

+e

• a

0.71

• b
• e

+a 0.001

• b
• e
• a

0.999 A J P(J|A) +a +j 0.9 +a

• j

0.1

• a

+j 0.05

• a
• j

0.95 A M P(M|A) +a +m 0.7 +a

• m

0.3

• a

+m 0.01

• a
• m

0.99

B E A M J

Example: Alarm Network

B P(B) +b 0.001

• b

0.999 E P(E) +e 0.002

• e

0.998 B E A P(A|B,E) +b +e +a 0.95 +b +e

• a

0.05 +b

• e

+a 0.94 +b

• e
• a

0.06

• b

+e +a 0.29

• b

+e

• a

0.71

• b
• e

+a 0.001

• b
• e
• a

0.999 A J P(J|A) +a +j 0.9 +a

• j

0.1

• a

+j 0.05

• a
• j

0.95 A M P(M|A) +a +m 0.7 +a

• m

0.3

• a

+m 0.01

• a
• m

0.99

B E A M J

Example: Alarm Network

B P(B) +b 0.001

• b

0.999 E P(E) +e 0.002

• e

0.998 B E A P(A|B,E) +b +e +a 0.95 +b +e

• a

0.05 +b

• e

+a 0.94 +b

• e
• a

0.06

• b

+e +a 0.29

• b

+e

• a

0.71

• b
• e

+a 0.001

• b
• e
• a

0.999 A J P(J|A) +a +j 0.9 +a

• j

0.1

• a

+j 0.05

• a
• j

0.95 A M P(M|A) +a +m 0.7 +a

• m

0.3

• a

+m 0.01

• a
• m

0.99

B E A M J

Bayes’ Nets

• Representation
• Conditional Independences
• Probabilistic Inference
• Enumeration (exact, exponential

complexity)

• Variable elimination (exact, worst-case

exponential complexity, often better)

• Inference is NP-complete
• Sampling (approximate)
• Learning Bayes’ Nets from Data

SLIDE 2

• Exa

Examples: s:

• Po

Posterior probability

• Most like

kely explanation

Inference

• In

Inference: calculating some useful quantity from a joint probability distribution

Inference by Enumeration in Bayes’ Net

• Gi

Given unlimited d time, inference in BNs Ns is easy

• Na

Naïve strategy: In Infer eren ence ce by en enumer erat ation

B E A M J P(B | + j, +m) ∝B P(B, +j, +m)

= X

e,a

P(B, e, a, +j, +m) = X

e,a

P(B)P(e)P(a|B, e)P(+j|a)P(+m|a)

=P(B)P(+e)P(+a|B, +e)P(+j| + a)P(+m| + a) + P(B)P(+e)P(−a|B, +e)P(+j| − a)P(+m| − a) P(B)P(−e)P(+a|B, −e)P(+j| + a)P(+m| + a) + P(B)P(−e)P(−a|B, −e)P(+j| − a)P(+m| − a)

J M

capital B means: compute for all values b

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

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Inference by Enumeration

• Ge

General l case:

• Ev

Evidence vari riables:

• Qu

Quer ery* var variable: iable:

• Hi

Hidden vari riables: Al All var variab ables es

* Works ks fine with multiple qu query ry vari riabl bles, too

• We

We want:

• St

Step p 1: Se Select the entries consistent with the evidence

• St

Step p 2: Su Sum out H to get joint

• f Query and evidence
• St

Step p 3: No Normaliz lize

× 1 Z

Inference by Enumeration? Inference by Enumeration vs. Variable Elimination

• Why is inference by enumeration so slow?
• You join up the whole joint distribution before

you sum out the hidden variables

• Id

Idea: ea: inter erleav eave e joining an and mar arginal alizing!

• Called “Va

Variable Elimination”

• Still NP-hard, but usually much faster

than inference by enumeration

• First we’ll need some new notation: fa

factors rs

Factor Zoo

Go Goal: Let’s make a ta taxon

• nom
• my of

co conditional al probab ability tab ables es (we have seen most of these before)

Factor Zoo I

• Jo

Joint dist stribution: P( P(X,Y) X,Y)

• Entries P(

P(x, x,y) for all x, y

• Sums to 1
• Se

Selecte ted j d joint: t: P( P(x, x,Y)

• A slice of the joint distribution
• Entries P(

P(x, x,y) for fixe xed x, al all y

• Sums to P(x)

x) (usually not 1)

• Number of capitals =

number of dimensions in table

T W P hot sun 0.4 hot rain 0.1 cold sun 0.2 cold rain 0.3 T W P cold sun 0.2 cold rain 0.3

Factor Zoo II

• Si

Single conditional: P( P(Y | x)

• Entries P(

P(y | x) for fi fixed x, al all y

• Sums to 1
• Fa

Family of cond nditiona nals: P( P(Y | X)

• Multiple conditionals
• Entries P(

P(y | x) for al all x, y

• Sums to |X

|X| (size of domain)

T W P hot sun 0.8 hot rain 0.2 cold sun 0.4 cold rain 0.6 T W P cold sun 0.4 cold rain 0.6

SLIDE 3

Factor Zoo III

• Specified family: P(

P( y y | X )

• Entries P(

P(y | x) for fi fixed y, bu but for all x

• Sums to … who knows!

T W P hot rain 0.2 cold rain 0.6

Factor Zoo Summary

• In

In gen ener eral al, when en we e write e P( P(Y1 … … YN | | X1 … … XM)

• This is a fa

factor, a multi-dimensional array containing numbers ≥ 0

• Its values are P(

P(y1 … … yN | x | x1 … … xM)

• Any assi

assigned ed (=lower-case) X or Y is a dimension missing (selected) from the array

Example: Traffic Domain

• Ra

Random Varia iable les

• R: Ra

Raining

• T: Traf

Traffic

• L: Late for class!

ss!

T L R

+r 0.1

• r

0.9 +r +t 0.8 +r

• t

0.2

• r

+t 0.1

• r
• t

0.9 +t +l 0.3 +t

• l

0.7

• t

+l 0.1

• t
• l

0.9

P(L) = ?

= X

r,t

P(r, t, L) = X

r,t

P(r)P(t|r)P(L|t)

Inference by Enumeration: Procedural Outline

• Track

k all objects s (fa facto tors)

• Initial factors

s are lo local l CPTs s (one per node)

• Any

y kn known va values s are se selected

• E.g. if we know , the initial factors are
• Pr

Procedu dure: Jo Join all factors, el eliminat nate all hidden variables, normalize ze

+r 0.1

• r

0.9 +r +t 0.8 +r

• t

0.2

• r

+t 0.1

• r
• t

0.9 +t +l 0.3 +t

• l

0.7

• t

+l 0.1

• t
• l

0.9 +t +l 0.3

• t

+l 0.1 +r 0.1

• r

0.9 +r +t 0.8 +r

• t

0.2

• r

+t 0.1

• r
• t

0.9

Operation 1: Join Factors

• First

st basi sic operation: jo join inin ing fa facto tors

• Combining factors:

s:

• Ju

Just st like ke a database se join

• Get all factors over the joining variable
• Build a new factor over the union
• f the variables involved
• Exa

xample: Join on R

• Computation for each entry:

y: pointwise products +r 0.1

• r

0.9 +r +t 0.8 +r

• t

0.2

• r

+t 0.1

• r
• t

0.9 +r +t 0.08 +r

• t

0.02

• r

+t 0.09

• r
• t

0.81

T R R,T

Example: Multiple Joins Example: Multiple Joins

T R

Join R

L R, T L

+r 0.1

• r

0.9 +r +t 0.8 +r

• t 0.2
• r

+t 0.1

• r
• t 0.9

+t +l 0.3 +t

• l

0.7

• t

+l 0.1

• t
• l

0.9 +r +t 0.08 +r

• t

0.02

• r

+t 0.09

• r
• t

0.81 +t +l 0.3 +t

• l

0.7

• t

+l 0.1

• t
• l

0.9

R, T, L

+r +t +l

0.024

+r +t

• l

0.056

+r

• t

+l

0.002

+r

• t
• l

0.018

• r

+t +l

0.027

• r

+t

• l

0.063

• r
• t

+l

0.081

• r
• t
• l

0.729

Join T

O(2^2) O(2^3)

Operation 2: Eliminate

• Second basi

sic operation: marginaliza zation

• Take

ke a factor and su sum out a va variable

• Shrinks

ks a factor to a smaller one

• A pr

projection operation

• Exa

xample:

+r +t 0.08 +r

• t

0.02

• r

+t 0.09

• r
• t

0.81 +t 0.17

• t

0.83

SLIDE 4

Multiple Elimination

Sum

• ut R

Sum

• ut T

T, L L R, T, L

+r +t +l

0.024

+r +t

• l

0.056

+r

• t

+l

0.002

+r

• t
• l

0.018

• r

+t +l

0.027

• r

+t

• l

0.063

• r
• t

+l

0.081

• r
• t
• l

0.729 +t +l

0.051

+t

• l

0.119

• t

+l

0.083

• t
• l

0.747

+l 0.134

• l

0.886

Thus Far: Multiple Join, Multiple Eliminate (= Inference by Enumeration)

Marginalizing Early (= Variable Elimination) Traffic Domain

Inference by y Enumeration

T L R

P(L) = ?

Va Variable Elimination

= X

t

P(L|t) X

r

P(r)P(t|r)

= X

t

X

r

P(L|t)P(r)P(t|r)

Join on r Join on t Eliminate r Eliminate t Join on r Eliminate r Join on t Eliminate t

Marginalizing Early! (aka VE)

Sum out R

T L

+r +t 0.08 +r

• t

0.02

• r

+t 0.09

• r
• t

0.81 +t +l 0.3 +t

• l

0.7

• t

+l 0.1

• t
• l

0.9 +t 0.17

• t

0.83 +t +l 0.3 +t

• l

0.7

• t

+l 0.1

• t
• l

0.9

T R L

+r 0.1

• r

0.9 +r +t 0.8 +r

• t 0.2
• r

+t 0.1

• r
• t 0.9

+t +l 0.3 +t

• l

0.7

• t

+l 0.1

• t
• l

0.9 Join R

R, T L T, L L

+t +l

0.051

+t

• l

0.119

• t

+l

0.083

• t
• l

0.747

+l 0.134

• l

0.866 Join T Sum out T

O(2^2) O(2^2) O(2^2) O(2^2)

Evidence

• If evidence, start with factors that select that evidence
• No evidence uses these initial factors:
• Computing , the initial factors become:
• We eliminate all vars other than query + evidence

+r 0.1

• r

0.9 +r +t 0.8 +r

• t

0.2

• r

+t 0.1

• r
• t

0.9 +t +l 0.3 +t

• l

0.7

• t

+l 0.1

• t
• l

0.9 +r 0.1 +r +t 0.8 +r

• t

0.2 +t +l 0.3 +t

• l

0.7

• t

+l 0.1

• t
• l

0.9

Evidence II

• Re

Resul ult will be a se selected j joint of

• f qu

query an and ev eviden ence ce

• E.g. for P(L | +r

+r), we would end up with:

• To get our answer, just no

normalize this!

• That’s it!

+l 0.26

• l

0.74 +r +l 0.026 +r

• l

0.074 Normalize

General Variable Elimination

• Query:

y:

• Start with initial factors:

s:

• Local CPTs (but instantiated by evidence)
• While there are st

still hidden va variables s (not Q Q or evi vidence):

• Pick

k a hidden variable H

• Jo

Join all factors mentioning H

• El

Eliminate (sum out) H

• Jo

Join all remaining factors and normalize ze

SLIDE 5

Example

Cho Choose A

Example

Choose E Finish with B

Normalize

Same Example in Equations

define marginal in terms of sum decompose joint probability for Bayes’ net use x*(y+z) = xy + xz joining on a, and then summing out gives f1 use x*(y+z) = xy + xz joining on e, and then summing out gives f2

All we are doing is exploiting uwy + uwz + uxy + uxz + vwy + vwz + vxy +vxz = (u+v)(w+x)(y+z) to improve computational efficiency!

Exercise: Variable Elimination Ordering

• Suppose

se we have ve the query: y: P( P(Xn|y |y1,…, …,yn) )

• Compare two possi

ssible or

• rder

ering ngs: 1.

• 1. Z,

Z, X1, …, …, Xn-1 2.

• 2. X1, …,

…, Xn-1, Z , Z

• Answ

swer: O( O(2n+

n+1)

) versus O( O(n 2 22) (assuming binary)

• In

In g general: l: the ordering can greatly affect efficiency.

… …

VE: Computational and Space Complexity

• The

The com comput utat ational

• nal and

and sp space complexi xity y of va variable elimination are determined by y the largest st factor

• The

The el eliminat nation

• n or
• rder

ering ng can greatly y affect the si size ze of the largest st factor.

• E.g., previous slide’s example 2n vs. 2
• Does

s there always ys exi xist st an ordering that only y resu sults s in sm small factors? s?

• No

No! (everything in AI is NP hard)

Polytrees

• A

A polyt ytree is s a directed graph with no undirected cyc ycles

• For

For pol

• ly-trees

s yo you can always ys find an ordering that is s efficient

• Try it!!
• Cu

Cut-se set conditioning for Baye yes’ net net inf nfer erence ence

• Choose se

set of va variables s such that if removed only a polyt ytree re rema mains

• Exe

xercise se: Think about how the specifics would work out!

Bayes’ Nets

• Representation
• Conditional Independences
• Probabilistic Inference
• Enumeration (exact, exponential

complexity)

• Variable elimination (exact, worst-case

exponential complexity, often better)

• Inference is NP-complete
• Sampling (approximate)
• Learning Bayes’ Nets from Data