[PDF] - Out line Wrap up d-separ at ion I nf erence in Bayes Net s Bayes PDF Document

SLIDE 1

1 Bayes Net s (cont )

CS 486/ 686 Univer sit y of Wat erloo May 31, 2005

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Out line

Wrap up d-separ at ion
I nf erence in Bayes Net s
Variable Eliminat ion

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D-Separat ion: I nt uit ions

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D-Separat ion: I nt uit ions

Subway and Therm are dependent ; but are independent

given Flu (since Flu blocks t he only pat h)

Aches and Fever are dependent ; but are independent

given Flu (since Flu blocks t he only pat h). Similarly f or Aches and Therm (dependent , but indep. given Flu).

Flu and Mal are indep. (given no evidence): Fever blocks

t he pat h, since it is not in evidence, nor is it s descendant

Therm. Flu,Mal are dependent given Fever (or given

Therm): not hing blocks pat h now.

Subway,Exot icTrip are indep.; t hey are dependent given

Therm; t hey are indep. given Therm and Malaria. This f or exact ly t he same reasons f or Flu/ Mal above.

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I nf erence in Bayes Net s

The independence sanct ioned by D-

separat ion (and ot her met hods) allows us t o comput e prior and post erior probabilit ies quit e ef f ect ively.

We' ll look at a couple simple examples

t o illust rat e. We' ll f ocus on net works wit hout loops. (A loop is a cycle in t he underlying undir ect ed gr aph. Recall t he dir ect ed gr aph has no cycles.)

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Simple Forwar d I nf erence (Chain)

Comput ing prior require simple f orward

“propagat ion” of probabilit ies

Not e: all (f inal) t erms are CPTs in t he BN

Not e: only ancest ors of J considered P(J )=ΣM,ET P(J |M,ET)P(M,ET)

(marginalizat ion)

P(J )=ΣM,ET P(J |M)P(M|ET)P(ET)

(chain rule and independence)

P(J )=ΣMP(J |M)ΣETP(M|ET)P(ET)

(dist ribut ion of sum)

SLIDE 2

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Simple For war d I nf erence (Chain)

Same idea applies when we have

“upst ream” evidence

P(J |ET) = ΣMP(J |M,ET) P(M|ET) = ΣM P(J |M) P(M|ET)

(J is cond independent of ET given M)

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Simple Forwar d I nf erence (Pooling)

Same idea applies wit h mult iple parent s

P(Fev) = ΣFlu,M P(Fev| Flu,M) P(Flu,M) = ΣFlu,M P(Fev| Flu,M) P(Flu) P(M) = ΣFlu,M P(Fev| Flu,M) ΣTS P(Flu| TS) P(TS)ΣET P(M| ET) P(ET)

(1) f ollows by summing out rule; (2) by

independence of Flu, M; (3) by summing out

– not e: all t erms are CPTs in t he Bayes net

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Simple Forwar d I nf erence (Pooling)

Same idea applies wit h evidence

P(Fev| t s,~M) = ΣFlu P(Fev | Flu,t s,~M) P(Flu| t s,~M)

= ΣFlu P(Fev| Flu,~M) P(Flu| t s)

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Simple Backward I nf erence

When evidence is downst ream of query

variable, we must reason “backwar ds.” This requires t he use of Bayes r ule: P(ET | j ) = α P(j | ET) P(ET) = α ΣM P(j | M,ET) P(M| ET) P(ET) = α ΣM P(j | M) P(M| ET) P(ET)

First st ep is j ust Bayes rule

– normalizing const ant α is 1/ P(j ); but we needn’t comput e it explicit ly if we comput e P(ET | j ) f or each value of ET: we j ust add up t erms P(j | ET) P(ET) f or all values

f ET (t hey sum t o P

(j ))

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Backward I nf erence (Pooling)

Same ideas when several pieces of

evidence lie “downst ream” P(ET | j ,f ev) =α P(j ,f ev | ET) P(ET) = α ΣM P(j ,f ev | M,ET) P(M| ET) P(ET) = α ΣM P(j ,f ev | M) P(M| ET) P(ET) = α ΣM P(j | M)P(f ev| M)P(M| ET) P(ET)

– Same st eps as bef ore; but now we comput e prob of bot h pieces of evidence given hypot hesis ET and combine t hem. Not e: t hey are independent given M; but not given ET. – St ill must simplif y P (f ev|M) down t o CPTs (as usual)

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Variable Eliminat ion

The int uit ions in t he above examples give us

a simple inf erence algorit hm f or net wor ks wit hout loops: t he polyt ree algorit hm.

I nst ead we' ll look at a mor e gener al

algorit hm t hat works f or general BNs; but t he polyt ree algor it hm will be a special case.

The algorit hm, variable eliminat ion, simply

applies t he summing out rule repeat edly.

– To keep comput at ion simple, it exploit s t he independence in t he net work and t he abilit y t o dist ribut e sums inward

SLIDE 3

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Fact ors

A f unct ion f (X1, X2,…

, Xk) is also called a f act or. We can view t his as a t able of number s, one f or each inst ant iat ion of t he variables X1, X2,… , Xk.

– A t abular rep’n of a f act or is exponent ial in k

Each CPT in a Bayes net is a f act or:

– e.g., Pr(C| A,B) is a f unct ion of t hree variables, A, B, C

Not at ion: f (X,Y) denot es a f act or over t he

variables X ∪ Y. (Here X, Y are set s of variables.)

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The Product of Two Fact ors

Let f (X,Y) & g(Y,Z) be t wo f act ors wit h

variables Y in common

The product of f and g, denot ed h = f x g

(or somet imes j ust h = f g), is def ined: h(X,Y,Z) = f (X,Y) x g(Y,Z)

0.12 ~a~b~c 0.48 ~a~bc 0.2 ~b~c 0.6 ~a~b 0.12 ~ab~c 0.28 ~abc 0.8 ~bc 0.4 ~ab 0.02 a~b~c 0.08 a~bc 0.3 b~c 0.1 a~b 0.27 ab~c 0.63 abc 0.7 bc 0.9 ab

h(A,B,C) g(B,C) f (A,B)

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Summing a Var iable Out of a Fact or

Let f (X,Y) be a f act or wit h var iable X (Y

is a set )

We sum out var iable X f rom f t o produce

a new f act or h = ΣX f , which is def ined: h(Y) = Σx∊Dom(X) f (x,Y)

0.6 ~a~b 0.4 ~ab 0.7 ~b 0.1 a~b 1.3 b 0.9 ab

h(B) f (A,B)

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Rest rict ing a Fact or

Let f (X,Y) be a f act or wit h var iable X (Y

is a set )

We rest rict f act or f t o X=x by set t ing X

t o t he value x and “delet ing”. Def ine h = f X=x as: h(Y) = f (x,Y)

0.6 ~a~b 0.4 ~ab 0.1 ~b 0.1 a~b 0.9 b 0.9 ab

h(B) = f A=a f (A,B)

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Variable Eliminat ion: No Evidence

Comput ing pr ior probabilit y of query var X

can be seen as applying t hese operat ions on f act ors

P(C) = ΣA,B P

(C|B) P(B| A) P(A) = ΣB P(C|B) ΣA P (B| A) P(A) = ΣB f 3(B,C) ΣA f 2(A,B) f 1(A) = ΣB f 3(B,C) f 4(B) = f 5(C) Def ine new f act ors: f 4(B)= ΣA f 2(A,B) f 1(A) and f 5(C)= ΣB f 3(B,C) f 4(B) B C A

f 1(A) f 2(A,B) f 3(B,C)

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Variable Eliminat ion: No Evidence

Here’s t he example wit h some numbers

B C A

f 1(A) f 2(A,B) f 3(B,C)

~c c

f 5(C)

0.375 0.625 ~b b

f 4(B)

0.15 0.85 0.1 0.9 ~a a

f 1(A)

0.8 ~b~c 0.6 ~a~b 0.2 ~bc 0.4 ~ab 0.3 b~c 0.1 a~b 0.7 bc 0.9 ab

f 3(B,C) f 2(A,B)

SLIDE 4

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VE: No Evidence (Example 2)

P(D) = ΣA,B,C P(D| C) P(C| B,A) P(B) P(A) = ΣC P(D| C) ΣB P(B) ΣA P(C| B,A) P(A) = ΣC f 4(C,D) ΣB f 2(B) ΣA f 3(A,B,C) f 1(A) = ΣC f 4(C,D) ΣB f 2(B) f 5(B,C) = ΣC f 4(C,D) f 6(C) = f 7(D)

Def ine new f act ors: f 5(B,C), f 6(C), f 7(D), in t he obvious way C D A

f 1(A) f 3(A,B,C) f 4(C,D)

B

f 2(B)

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Variable Eliminat ion: One View

One way t o t hink of variable eliminat ion:

– writ e out desired comput at ion using t he chain rule, exploit ing t he independence relat ions in t he net work – arr ange t he t erms in a convenient f ashion – dist ribut e each sum (over each variable) in as f ar as it will go

i.e., t he sum over variable X can be “pushed in” as

f ar as t he “f irst ” f act or ment ioning X

– apply oper at ions “inside out ”, r epeat edly eliminat ing and creat ing new f act ors (not e t hat each st ep/ removal of a sum eliminat es

ne variable)

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Variable Eliminat ion Algorit hm

Given query var Q, remaining var s Z. Let

F be set of f act or s corresponding t o CPTs f or {Q} ∪ Z.

1. Choose an elimination ordering Z1, …, Zn of variables in Z.
2. For each Zj -- in the order given -- eliminate Zj ∊ Z

as follows: (a) Compute new factor gj = ΣZj f1 x f2 x … x fk, where the fi are the factors in F that include Zj (b) Remove the factors fi (that mention Zj ) from F and add new factor gj to F

3. The remaining factors refer only to the query variable Q.

Take their product and normalize to produce P(Q)

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VE: Example 2 again

St ep 1: Add f 5(B,C) = ΣA f 3(A,B,C) f 1(A) Remove: f 1(A), f 3(A,B,C) St ep 2: Add f 6(C)= ΣB f 2(B) f 5(B,C) Remove: f 2(B) , f 5(B,C) St ep 3: Add f 7(D) = ΣC f 4(C,D) f 6(C) Remove: f 4(C,D), f 6(C) Last f act or f 7(D) is (possibly unnormalized) probabilit y P(D) Factors: f 1(A) f 2(B) f 3(A,B,C) f 4(C,D) Query: P(D)?

Elim. Order: A, B, C

C D A

f 1(A) f 3(A,B,C) f 4(C,D)

B

f 2(B)

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Variable Eliminat ion: Evidence

Comput ing post erior of query variable given

evidence is similar; suppose we observe C=c: P(A| c) = α P(A) P(c| A) = α P(A) ΣB P(c| B) P(B| A) = α f 1(A) ΣB f 3(B,c) f 2(A,B) = α f 1(A) ΣB f 4(B) f 2(A,B) = α f 1(A) f 5(A) = α f 6(A)

New f act ors: f 4(B)= f 3(B,c); f 5(A)= ΣB f 2(A,B) f 4(B); f 6(A)= f 1(A) f 5(A) B C A

f 1(A) f 2(A,B) f 3(B,C)

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Var iable Eliminat ion wit h Evidence

Given query var Q, evidence var s E (observed t o be e), remaining var s Z. Let F be set of f act or s involving CPTs f or {Q} ∪ Z.

1. Replace each factor f∊F that mentions a variable(s) in E

with its restriction fE=e (somewhat abusing notation)

2. Choose an elimination ordering Z1, …, Zn of variables in Z.
3. Run variable elimination as above.
4. The remaining factors refer only to the query variable Q.

Take their product and normalize to produce P(Q)

SLIDE 5

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VE: Example 2 again wit h Evidence

Rest rict ion: replace f 4(C,D) wit h f 5(C) = f 4(C,d) St ep 1: Add f 6(A,B)= ΣC f 5(C) f 3(A,B,C) Remove: f 3(A,B,C), f 5(C) St ep 2: Add f 7(A) = ΣB f 6(A,B) f 2(B) Remove: f 6(A,B), f 2(B) Last f act ors: f 7(A), f 1(A). The product f 1(A) x f 7(A) is (possibly unnormalized) post erior. So…P (A|d) = α f 1(A) x f 7(A). Factors: f 1(A) f 2(B) f 3(A,B,C) f 4(C,D) Query: P(A)? Evidence: D = d

Elim. Order: C, B

C D A

f 1(A) f 3(A,B,C) f 4(C,D)

B

f 2(B)

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Some Not es on t he VE Algorit hm

Af t er it erat ion j (eliminat ion of Zj), f act ors remaining in

set F ref er only t o variables Xj +1, …Zn and Q. No f act or ment ions an evidence variable E af t er t he init ial rest rict ion.

Number of it erat ions: linear in number of variables
Complexit y is linear in number of vars and exponent ial in

size of t he largest f act or. – Recall each f act or has exponent ial size in it s number of variables – Can' t do any bet t er t han size of BN (since it s original f act ors are part of t he f act or set ) – When we creat e new f act ors, we might make a set of variables larger.

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Some Not es on t he VE Algorit hm

The size of t he result ing f act ors is det ermined by

eliminat ion ordering! (We’ll see t his in det ail)

For polyt rees, easy t o f ind good ordering (e.g.,

work out side in).

For general BNs, somet imes good orderings exist ,

somet imes t hey don' t (t hen inf erence is exponent ial in number of vars).

– Simply f inding t he opt imal eliminat ion ordering f or general BNs is NP-hard. – I nf erence in general is NP-hard in general BNs

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Eliminat ion Ordering: Polyt rees

I nf erence is linear in size
f net work

– ordering: eliminat e only “singly-connect ed” nodes – e.g., in t his net work, eliminat e D, A, C, X1,… ; or eliminat e X1,…Xk, D, A, C;

r mix up…

– result : no f act or ever larger t han original CPTs – eliminat ing B bef ore t hese gives f act ors t hat include all of A,C, X1,…Xk !!!

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Ef f ect of Dif f er ent Or der ings

Suppose query variable

is D. Consider dif f erent orderings f or t his net work

– A,F,H,G,B,C,E:

good: why?

– E,C,A,B,G,H,F:

bad: why?
Which order ing

creat es smallest f act ors?

– eit her max size or t ot al – which creat es largest ?

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Relevance

Cert ain var iables have no impact on t he

query.

– I n ABC net work, comput ing Pr(A) wit h no evidence requires eliminat ion of B and C.

But when you sum out t hese vars, you comput e a

t rivial f act or (whose value are all ones); f or example:

eliminat ing C: f 4(B) = ΣC f 3(B,C) = ΣC Pr(C|B)
1 f or any value of B (e.g., Pr(c| b) + Pr(~c|b) = 1)
No need t o t hink about B or C f or t his

query

B C A

SLIDE 6

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Relevance: A Sound Approximat ion

Can rest rict at t ent ion t o relevant
variables. Given query Q, evidence E:

– Q is relevant – if any node Z is relevant , it s parent s are relevant – if E∊E is a descendent of a relevant node, t hen E is relevant

We can rest rict our at t ent ion t o t he

subnet work compr ising only relevant variables when evaluat ing a query Q

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Next Class

Probabilist ic reasoning over t ime

– Dynamic Bayesian Net works – Hidden Markov Models

Russell & Norvig: Chapt er 15
Lect ure on t he board (no slides)